Herron Osorio - Tópicos Previos a La Matemática Superior

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5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

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⇒

(a ∗ a ) R (b ∗ b ).

&KFNQMP  &O FM &KFNQMP  MB PQFSBDJÓO i ∪ u Z MB SFMBDJÓO EF JHVBM EBE i = u TPO DPNQBUJCMFT QVFTUP RVF TJ A = B Z A = B  FOUPODFT A ∪ A = B ∪ B  . 5BNCJÊO TPO DPNQBUJCMFT MB JOUFSTFDDJÓO i ∩ u Z MB JHVBMEBE i = u 0CTFSWBDJÓO $PO FM ÃOJNP EF JS JOUSPEVDJFOEP BM MFDUPS B DVSTPT QPT UFSJPSFT FO ÃMHFCSB BCTUSBDUB DPODSFUBNFOUF FO UFPSÎB EF HSVQPT VOB DPTB FT FM DPOKVOUP A Z PUSB MB PQFSBDJÓO CJOBSJB ∗ EFàOJEB FO ÊM 1FSP KVOUPT FM QBS (A, ∗) EFàOF VOB FTUSVDUVSB BMHFCSBJDB MMBNBEB HSVQPJEF "IPSB CJFO DVBOEP FO VO HSVQPJEF MB PQFSBDJÓO CJOBSJB FT BTPDJBUJWB FTUF TF MMBNB TFNJHSVQP Z TJ BEFNÃT UJFOF FMFNFOUP OFVUSP Z FT JO WFSUJWB MB FTUSVDUVSB TF MMBNB HSVQP 'JOBMNFOUF FM OPNCSF EF HSVQP BCFMJBOP P DPONVUBUJWP TF MF BTJHOB BM HSVQP DVZB PQFSBDJÓO CJOBSJB FT DPONVUBUJWB

 &KFSDJDJPT  4FBO X VO DPOKVOUP àKP OP WBDÎP Z A FM DPOKVOUP EF UPEBT MBT GVODJPOFT CJZFDUJWBT EF X FO X. 1BSB f, g ∈ A, DPOTJEFSF MB DPN QPTJDJÓO EF GVODJPOFT f ◦ g y&T FTUB VOB PQFSBDJÓO CJOBSJB FO A 4J FT BTÎ yRVÊ QSPQJFEBEFT UJFOF FTUB PQFSBDJÓO 3FTQPOEB MBT NJTNBT QSFHVOUBT FO FM DBTP EF GVODJPOFT FO HFOFSBM  4FB (A, ∗) VO TFNJHSVQP DPO FMFNFOUP OFVUSP e 1SVFCF RVF TJ a ∈ A UJFOF JOWFSTP CJMBUFSBM FOUPODFT FTUF FT ÙOJDP  4FBO a, b Z c USFT FMFNFOUPT FO VO TFNJHSVQP (A, ∗) MPT DVBMFT UJFOFO JOWFSTPT 1SVFCF RVF a ∗ (b ∗ c) UJFOF JOWFSTP Z FTUF FT (c−1 ∗ b−1 ) ∗ a−1 .



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

 4FB ∗ VOB PQFSBDJÓO DPONVUBUJWB EFàOJEB FO VO DPOKVOUP A. .VFTUSF RVF VOB SFMBDJÓO EF FRVJWBMFODJB R FO A FT DPNQBUJCMF DPO MB PQFSBDJÓO ∗ TJ Z TPMP TJ (∀ x, y, z ∈ A) [x R y ⇒ (x ∗ z) R (y ∗ z)].  4FB (G, ·) VO HSVQP Z TFB a ∈ G VO FMFNFOUP àKP %FàOB VOB OVFWB PQFSBDJÓO ∗ FO FM DPOKVOUP G BTÎ x ∗ y = x · a · y QBSB UPEP x, y ∈ G. 1SVFCF RVF (G, ∗) FT VO HSVQP JTPNPSGP B (G, ·). 6OB GVODJÓO f : G → H FOUSF EPT HSVQPT (G, ∗) Z (H, ◦) TF MMBNB JTPNPSàTNP TJ f (a ∗ b) = f (a) ◦ f (b) QBSB UPEP a, b ∈ G Z BEFNÃT f FT CJZFDUJWB &O FTUF DBTP MPT HSVQPT TF MMBNBO JTPNPSGPT   4FB (G, ∗) VO HSVQP UBM RVF QBSB UPEP QBS EF FMFNFOUPT a Z b TF DVNQMF RVF (a ∗ b)−1 = a−1 ∗ b−1 . %FNVFTUSF RVF ∗ FT VOB PQFSBDJÓO CJOBSJB DPONVUBUJWB  4FBO (G, ◦) Z (H, ∗) HSVQPT 4J FO G × H TF EFàOF (g1 , h1 )  (g2 , h2 ) = (g1 ◦ g2 , h1 ∗ h2 ), QSVFCF RVF (G×H, ) FT VO HSVQP &TUF HSVQP TF MMBNB FM QSPEVDUP EJSFDUP EF (G, ◦) Z (H, ∗)  4FBO (G, ∗) VO HSVQP Z g ∈ G VO FMFNFOUP àKP 4J TF EFàOF f : G → G QPS f (x) = g −1 ∗ x ∗ g QSVFCF RVF f FT VO JTPNPSàTNP  4FBO (G, ∗) VO HSVQP BCFMJBOP Z f : G → G EBEB QPS f (x) = x−1  1SVFCF RVF f FT VO JTPNPSàTNP  4FB (G, ∗) VO HSVQP Z TVQPOHB RVF f : G → G EFàOJEB QPS f (x) = x−1 FT VO IPNPNPSàTNP 1SVFCF RVF (G, ∗) FT VO HSVQP BCFMJBOP 6O IPNPNPSàTNP EF VO HSVQP (G, ∗) FO PUSP HSVQP (K, ◦) FT VOB GVODJÓO φ : G → K DPO MB QSPQJFEBE φ(g1 ∗ g2 ) = φ(g1 ) ◦ φ(g2 ) QBSB DBEB g1 , g2 ∈ G   4FB (G, ∗) VO HSVQP UBM QBSB UPEP a ∈ G TF DVNQMF RVF a ∗ a = e EPOEF e FT FM FMFNFOUP OFVUSP %FNVFTUSF RVF G FT BCFMJBOP



$BQÎUVMP  -PT OÙNFSPT OBUVSBMFT

&YJTUFO WBSJBT NBOFSBT EF QSFTFOUBS MPT OÙNFSPT OBUVSBMFT QPS FKFNQMP VTBOEP MB UFPSÎB EF DPOKVOUPT P DPOKVOUPT JOEVDUJWPT -B QSFTFOUBDJÓO RVF IBDFNPT BDÃ FT WÎB MPT BYJPNBT EF 1FBOP  &TUPT QPTUVMBEPT QFSNJ UFO EFSJWBS UPEB MB BSJUNÊUJDB EF MPT OÙNFSPT OBUVSBMFT &O MB GPSNVMB DJÓO EF MPT BYJPNBT EF 1FBOP TF TVQPOF EF BOUFNBOP MB FYJTUFODJB EFM DPOKVOUP N

 $POTUSVDDJÓO BYJPNÃUJDB EF N %FàOJDJÓO  &M DPOKVOUP EF MPT OÙNFSPT OBUVSBMFT EFOPUBEP QPS N TF DBSBDUFSJ[B QPS MPT TJHVJFOUFT BYJPNBT MMBNBEPT BYJPNBT EF 1FBOP "1 &YJTUF VO FMFNFOUP 0 ∈ N MMBNBEP FM DFSP "1 1BSB DBEB n ∈ N FYJTUF VO ÙOJDP n+ ∈ N MMBNBEP FM TVDFTPS EF n. "1 1BSB DBEB n ∈ N, n+ = 0. "1 4J m, n ∈ N TPO UBMFT RVF m+ = n+ FOUPODFT m = n. "1 1SJODJQJP EF JOEVDDJÓO TJ S ⊆ N FT UBM RVF J 0 ∈ S Z JJ

n ∈ S ⇒ n+ ∈ S FOUPODFT S = N. 0CTFSWBDJÓO &M QSJNFS BYJPNB EJDF FO QBSUJDVMBS RVF FM DPOKVOUP N FT OP WBDÎP $PNCJOÃOEPMP DPO "1 EJDF RVF N UJFOF VO FMFNFOUP FM DFSP RVF OP FT TVDFTPS EF BMHÙO PUSP OBUVSBM BTÎ RVF FM DFSP FT FM iQSJNFSu FMFNFOUP -PT BYJPNBT "1 Z "1 EJDFO RVF DBEB OBUVSBM UJFOF VO ÙOJDP TVDFTPS Z RVF EPT OBUVSBMFT EJTUJOUPT UJFOFO TVDFTPSFT



(JVTFQQF 1FBOP  GVF VO NBUFNÃUJDP Z MÓHJDP JUBMJBOP



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

EJTUJOUPT 0USB NBOFSB EF FOUFOEFS FM BYJPNB "1 FT MB TJHVJFOUF TJ EFàOJNPT MB GVODJÓO S : N → N QPS S(n) = n+ MB GVODJÓO TVDFTPS EF n FOUPODFT FTUB GVODJÓO FT JOZFDUJWB 'JOBMNFOUF FM BYJPNB "1 FT EF NVDIB JNQPSUBODJB QVFT DPOTUJUVZF VOB IFSSBNJFOUB QPEFSPTB QBSB QSPCBS NVDIBT QSPQJFEBEFT FO FM DPOKVOUP N. &O UÊSNJOPT TJNQMFT FM DPOKVOUP EF MPT OÙNFSPT OBUVSBMFT DPOTJTUF EF VO FMFNFOUP EJTUJOHVJEP FM DFSP Z MB GVODJÓO TVDFTPSB EF n RVF TBUJT GBDFO MPT QPTUVMBEPT EF 1FBOP .VDIPT BVUPSFT QSFàFSFO RVF FM FMFNFOUP EJTUJOHVJEP TFB FM OBUVSBM  NBUFNÃUJDBNFOUF FTUP FT JSSFMFWBOUF 'JOBMNFOUF OP QSFTFOUBSFNPT BMVTJÓO QSPGVOEB BOUF FM BTVOUP SFMB DJPOBEP DPO MB FYJTUFODJB Z MB VOJDJEBE EF MPT OÙNFSPT OBUVSBMFT &O MB SFGFSFODJB q. 7FBNPT RVF FM QSJNFSP OP QVFEF PDVSSJS TJ BTÎ GVFTF FYJTUJSÎB VO OBUVSBM j UBM RVF p + j + = q. 4VTUJUVZFOEP FO  MMFHBSÎBNPT B RVF n(j + + 1) = m(j + + 1) MP DVBM QPS FM &KFSDJDJP  JNQMJDBSÎB RVF m = n &T EFDJS [(m, n)] = [(0, 0)] QBSB UPEP FOUFSP [(m, n)], RVF FT JNQPTJCMF &TUB DPOUSBEJDDJÓO QSVFCB RVF MB ÙOJDB QPTJCJMJEBE FT p > q. 7FBNPT RVF OFDFTBSJBNFOUF p = q +1. &O FGFDUP FYJTUF k ∈ N UBM RVF q + k + = p Z BM SFFNQMB[BS FO 

PCUFOFNPT RVF mk + + n = nk + + m EF EPOEF TF TJHVF RVF mk = nk



-PT OÙNFSPT FOUFSPT

-VFHP k = 0 QVFT EF MP DPOUSBSJP UFOESÎBNPT RVF m = n FT EFDJS [(m, n)] = [(0, 0)] QBSB UPEP FOUFSP [(m, n)]. )FNPT QSPCBEP BTÎ RVF p = q + 1 MP DVBM TJHOJàDB RVF [(p, q)] = [(q + 1, q)] = [(1, 0)] %F FTUB NBOFSB RVFEB EFNPTUSBEB MB VOJDJEBE EFM OFVUSP QBSB FM QSPEVDUP FO FM DPOKVOUP EF MPT OÙNFSPT FOUFSPT D /PUFNPT RVF EBEP [(m, n)] ∈ Z TF UJFOF RVF [(m, n)] + [(n, m)] = [(m + n, n + m)] = [(0, 0)]. -B VOJDJEBE FT TJNQMF EF QSPCBS E &T VOB DVFOUB EJSFDUB Z TJNQMF RVF JOWJUBNPT BM MFDUPS B SFBMJ[BSMB   &KFSDJDJP  .VFTUSF RVF MB TVNB FO MPT FOUFSPT TBUJTGBDF MB QSPQJFEBE DBODFMBUJWB [(m, n)] + [(p, q)] = [(m, n)] + [(r, s)] ⇒ [(p, q)] = [(r, s)]. 0CTFSWBDJÓO %F BDVFSEP DPO MB OPUBDJÓO FTUBCMFDJEB FO FM DBQÎUVMP  QBSB MPT JOWFSTPT BEJUJWPT WBNPT B EFOPUBS FM PQVFTUP EF VO FOUFSP [(m, n)] QPS −[(m, n)]. 1PS UBOUP BM DPNCJOBS FTUP DPO MB QBSUF D EFM UFPSFNB BOUFSJPS UFOFNPT RVF [(n, m)] = −[(m, n)]. -B JEFB B DPOUJOVBDJÓO FT MMFWBS MB FTDSJUVSB EF MPT OÙNFSPT FOUFSPT B VOB GPSNB ÃHJM TJO VTBS MB OPUBDJÓO EF DMBTFT EF FRVJWBMFODJB &O FTB EJSFDDJÓO MB TJHVJFOUF QSPQPTJDJÓO TFSÃ EF HSBO VUJMJEBE 1SPQPTJDJÓO  4FB [(p, q)] ∈ Z &YJTUF VO ÙOJDP OBUVSBM n UBM RVF VOB EF MBT TJHVJFOUFT JHVBMEBEFT TF DVNQMF [(p, q)] = [(n, 0)] P [(p, q)] = [(0, n)]. %FNPTUSBDJÓO $PNP p Z q TPO OBUVSBMFT MB MFZ EF USJDPUPNÎB HBSBOUJ[B RVF TPMBNFOUF VOB EF MBT TJHVJFOUFT QPTJCJMJEBEFT UJFOF MVHBS p = q, p < q, p > q. 4J PDVSSF MB QSJNFSB FOUPODFT DPO n = 0 TF DVNQMF MB QSPQPTJDJÓO &O FM TFHVOEP DBTP FYJTUF VO ÙOJDP OBUVSBM m UBM RVF p + m+ = q Z BTÎ [(p, q)] = [(p, p + m+ )] = [(0, m+ )], EF EPOEF TF UJFOF FM SFTVMUBEP UPNBOEP n = m+ . 'JOBMNFOUF TJ p > q FYJTUF VO ÙOJDP OBUVSBM k UBM RVF q + k + = p Z DPNP BOUFT [(p, q)] = [(q + k + , q)] = [(k + , 0)], Z FM   SFTVMUBEP TF PCUJFOF UPNBOEP n = k + .



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&OTFHVJEB WBNPT B QSPCBS RVF FM DPOKVOUP EF MPT OÙNFSPT OBUVSB MFT FT VO TVCDPOKVOUP QSPQJP EFM DPOKVOUP EF MPT OÙNFSPT FOUFSPT 1BSB FTUF QSPQÓTJUP WBNPT B OFDFTJUBS MB BZVEB EF VO DPODFQUP BMHFCSBJDP JTPNPSàTNP -P RVF FTUÃ EFUSÃT EF VO JTPNPSàTNP FT QSFDJTBS DVÃOEP DJFSUB DMBTF EF DPOKVOUPT TPO FTUSVDUVSBMNFOUF JHVBMFT JOEFQFOEJFOUF NFOUF EF RVF TVT FMFNFOUPT TFBO EJGFSFOUFT  6OB BQMJDBDJÓO f : X → Y FOUSF EPT DPOKVOUPT EPUBEPT EFM NJTNP UJQP EF FTUSVDUVSB FT VO JTPNPS àTNP DVBOEP DBEB FMFNFOUP EF Y QSPWJFOF EF VO ÙOJDP FMFNFOUP EF X Z f USBOTGPSNB MBT PQFSBDJPOFT SFMBDJPOFT FUD RVF UFOFNPT FO X FO MBT RVF UFOFNPT FO Y  NÃT QSFDJTBNFOUF MBT QSFTFSWB $VBOEP FOUSF EPT FTUSVDUVSBT IBZ VO JTPNPSàTNP BNCBT TPO JOEJTUJOHVJCMFT UJFOFO MBT NJTNBT QSPQJFEBEFT %FàOJDJÓO  4FBO X Z Y EPT DPOKVOUPT FO MPT DVBMFT IBZ EFàOJEBT PQFSBDJPOFT ∗ Z  SFTQFDUJWBNFOUF %FDJNPT RVF FTUPT DPOKVOUPT TPO JTPNPSGPT TJ FYJTUF VOB CJZFDDJÓO f : X → Y UBM RVF QBSB UPEP x, y ∈ X TF UJFOF RVF f (x∗y) = f (x)f (y). -B GVODJÓO f TF MMBNB VO JTPNPSàTNP $PO FM ÃOJNP EF QSPCBS RVF N ⊂ Z JOUSPEVDJNPT VO TVCDPOKVOUP EF Z EFàOJEP QPS  := {[(n, 0)] : n ∈ N} N  QPS J (n) = [(n, 0)]. 7FBNPT RVF J Z UBNCJÊO EFàOJNPT J : N → N FT CJZFDUJWB MB TPCSFZFDUJWJEBE FT JONFEJBUB 4VQPOHBNPT RVF J (n) = J (m) -VFHP (n, 0) ≈ (m, 0) Z FO DPOTFDVFODJB n = m. &TUP QSVFCB RVF J FT JOZFDUJWP Z QPS UBOUP FT CJZFDUJWP 'JOBMNFOUF WFBNPT RVF J QSFTFSWB MBT PQFSBDJPOFT TVNB Z QSPEVDUP EFàOJEBT FO  SFTQFDUJWBNFOUF NZN J (n + m) = [(n + m, 0)] = [(n, 0) ⊕ (m, 0)] = [(n, 0)] + [(m, 0)] = J (n) + J (m). J (n · m) = [(n · m, 0)] = [(n, 0)  (m, 0)] = [(n, 0)] · [(m, 0)] = J (n) · J (m).  QPS FMMP QP 2VFEB EFNPTUSBEP RVF J FT VO JTPNPSàTNP FOUSF N Z N EFNPT BàSNBS RVF FTUPT DPOKVOUPT TPO JEFOUJàDBCMFT JOEJTUJOHVJCMFT P  FT VO TVCDPOKVOUP QSPQJP EF RVF VOP FT VOB DPQJB EFM PUSP $PNP N Z FOUPODFT QPEFNPT DPODMVJS RVF N ⊂ Z. "EFNÃT QPS FM JTPNPSàTNP QPEFNPT JEFOUJàDBS FM OBUVSBM n DPO TV JNBHFO J (n) FO Z, FT EFDJS [(n, 0)] = n. "QSPWFDIBOEP FTUP Z UFOJFOEP FO DVFOUB RVF FM PQVFTUP EF [(p, 0)] EF OPUBEP QPS −[(p, 0)] FT [(0, p)] QPEFNPT BàSNBS QPS MB VOJDJEBE EFM



-PT OÙNFSPT FOUFSPT

JOWFSTP BEJUJWP RVF [(0, p)] = −p. "IPSB CJFO EBEP DVBMRVJFS FOUFSP [(p, q)], JOWPDBOEP MB 1SPQPTJDJÓO  FYJTUF VO ÙOJDP OBUVSBM n UBM RVF [(p, q)] = [(n, 0)] P CJFO FT JHVBM B [(0, n)] QFSP DBEB VOP EF FTUPT FT JEFOUJàDBEP DPO n P −n SFTQFDUJWBNFOUF "TÎ MBT DPTBT FM DPOKVOUP EF MPT OÙNFSPT FOUFSPT RVFEB EFTDSJUP EF NBOFSB TJNQMJàDBEB FO MB GPSNB DMÃTJDB DPNP TF MFT DPOPDF TJO DMBTFT EF FRVJWBMFODJB Z = {0, ±1, ±2, ±3, . . .}. )BTUB BIPSB FO FM DPOKVOUP EF MPT FOUFSPT IFNPT EFàOJEP EPT PQFSB DJPOFT MB TVNB Z FM QSPEVDUP $PO MB OPUBDJÓO EFM JOWFSTP QPEFNPT JOUSPEVDJS VOB UFSDFSB %FàOJDJÓO  %BEPT EPT FOUFSPT m Z n EFàOJNPT TV EJGFSFODJB EF OPUBEB QPS m − n DPNP m + (−n). *OUSPEVDJNPT MB TJHVJFOUF OPUBDJÓO N∗ := N  {0} 5FPSFNB  4FBO m, n Z k OÙNFSPT FOUFSPT &OUPODFT B &M PQVFTUP EFM PQVFTUP EF m FT m, FT EFDJS −(−m) = m. C 4F DVNQMF MB MFZ EF USJDPUPNÎB FO Z FTUP FT TPMBNFOUF VOB EF MBT TJHVJFOUFT BàSNBDJPOFT TF DVNQMF m = 0, m ∈ N∗ , −m ∈ N∗ . D m · 0 = 0. E 4F UJFOF MB MMBNBEB MFZ EF TJHOPT BMHFCSBJDPT m(−n) = −(mn) = (−m)n, (−m)(−n) = mn. F 4F DVNQMF MB QSPQJFEBE DBODFMBUJWB EFM QSPEVDUP TJ mn = mk Z m = 0 FOUPODFT n = k. G /P IBZ EJWJTPSFT EF DFSP FTUP FT TJ mn = 0 FOUPODFT m = 0 P n = 0. %FNPTUSBDJÓO 1PS DPNPEJEBE Z GPSUBMF[B FO MPT BSHVNFOUPT VTBSFNPT MB OPUBDJÓO EF DMBTF EF FRVJWBMFODJB QBSB VO FOUFSP B 1VFTUP RVF m + (−m) = 0 Z FM PQVFTUP FT ÙOJDP FTUB JHVBMEBE EJDF RVF m FT FM PQVFTUP EF −m FM DVBM FT EFOPUBEP QPS −(−m). &TUP EFNVFTUSB MB BàSNBDJÓO C 7FBNPT QSJNFSP RVF EPT QPTJCJMJEBEFT OP QVFEFO EBSTF TJNVM UÃOFBNFOUF MBT EPT QSJNFSBT PSJHJOBO VOB DPOUSBEJDDJÓO MB QSJNFSB Z



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

MB UFSDFSB UBNCJÊO PSJHJOBO VOB DPOUSBEJDDJÓO ZB RVF FO FTF DBTP UFO ESÎBNPT m = 0 Z −m = 0 Z DPNP N ⊂ Z QPS UBOUP −m, m ∈ Z "MMÎ m = 0 UJFOF DPNP ÙOJDP JOWFSTP B −m = 0 'JOBMNFOUF TJ PDVSSJFSBO TJNVMUÃOFBNFOUF MBT EPT ÙMUJNBT FOUPODFT UFOESÎBNPT RVF −m, m ∈ N Z UBOUP m DPNP −m TPO EJGFSFOUFT EF DFSP MP RVF JNQMJDBSÎB RVF FYJT UJSÎBO OBUVSBMFT p Z q UBMFT RVF m = p+ Z −m = q + . .JSBOEP FTUPT OBUVSBMFT DPNP FOUFSPT TF UFOESÎB RVF 0 = [(0, 0)] = m + (−m) = [(p+ , 0)] + [(q + , 0)] = [(p+ + q + , 0)] = p+ + q + = (p+ + q)+ . &TUP DPOUSBEJSÎB FM BYJPNB "1 EF 1FBOP "IPSB WFBNPT RVF BM NFOPT VOB EF MBT QPTJCJMJEBEFT TÎ PDVSSF TV QPOHBNPT RVF m = 0 Z QSPCFNPT RVF m ∈ N∗ ∨ −m ∈ N∗ . 1PS MB 1SPQPTJDJÓO  m FT EF MB GPSNB [(n, 0)] P CJFO m = [(0, n)] QBSB BMHÙO OBUVSBM n = 0. &O FM QSJNFS DBTP TF UJFOF JONFEJBUBNFOUF RVF m = n ∈ N∗  EFM TFHVOEP DBTP TF MMFHB B RVF m + [(n, 0)] = [(0, n)] + [(n, 0)] = [(0, 0)] = 0. -VFHP [(n, 0)] FT FM PQVFTUP EF m Z QPS VOJDJEBE EF FTUF PCUFOFNPT RVF −m = [(n, 0)] = n ∈ N∗ . D 4FB m = [(p, q)] ∈ Z. &OUPODFT [(p, q)] · [(0, 0)] = [(0, 0)]. E 1PS FM MJUFSBM BOUFSJPS Z MB QSPQJFEBE EJTUSJCVUJWB UFOFNPT RVF 0 = m · 0 = m(−n + n) = m(−n) + mn, EF MP DVBM TF TJHVF RVF m(−n) FT FM PQVFTUP EF mn Z QPS MB VOJ DJEBE EF FTUF FOUPODFT OFDFTBSJBNFOUF m(−n) = −(mn). %F NBOF SB TJNJMBS 0 = n · 0 = n(−m + m) = (−m)n + mn JNQMJDB RVF (−m)n = −(mn) $PNP FTUP FT WÃMJEP QBSB UPEP QBS EF FOUFSPT m Z n FOUPODFT DBNCJBOEP n QPS −n Z VTBEP OVFWBNFOUF B

UFOFNPT RVF (−m)(−n) = −[m(−n)] = −[−(mn)] = mn. F %FOPUFNPT m = [(p, q)], n = [(r, s)] Z k = [(u, v)]. -B IJQÓUFTJT OPT QSPQPSDJPOB MB JHVBMEBE [(pr + qs, ps + qr)] = [(pu + qv, pv + qu)], FTUP FT pr + qs + pv + qu = ps + qr + pu + qv. 

1VFTUP RVF m = 0 FOUPODFT p = q Z QPS UBOUP QPS 5SJDPUPNÎB FO MPT OBUVSBMFT TF UJFOF RVF p < q P q < p. &O MB QSJNFSB TJUVBDJÓO p+j + = q QBSB BMHÙO OBUVSBM j. 3FFNQMB[BOEP FO  TF PCUJFOF RVF j + (s + u) = j + (r + v),



-PT OÙNFSPT FOUFSPT

MP DVBM TJHOJàDB RVF [(r, s)] = [(u, v)] FT EFDJS n = k. &O FM TFHVOEP DBTP TF SB[POB EF NBOFSB TJNJMBS G 4VQPOHBNPT RVF m = 0. %BEP RVF MB IJQÓUFTJT TF QVFEF FTDSJCJS  mn = m · 0, FM MJUFSBM BOUFSJPS JNQMJDB RVF n = 0.  %FàOJDJÓO  4FBO m Z n FOUFSPT %FDJNPT RVF m < n TJ n−m ∈ N∗ Z EFDJNPT RVF m ≤ n TJ n − m ∈ N. -BT BàSNBDJPOFT m ≥ n Z m > n TJHOJàDBO n ≤ m Z n < m SFTQFDUJWBNFOUF 0CTFSWBDJPOFT J /PUFNPT RVF EF FTUB EFàOJDJÓO TF EFTQSFOEF RVF MB QSPQJFEBE EF USJDPUPNÎB FO Z QVFEF FOVODJBSTF EF NBOFSB FRVJWBMFOUF BTÎ EBEP VO FOUFSP m TPMBNFOUF VOB EF MBT TJHVJFOUFT BTFWFSBDJPOFT TF DVNQMF m = 0, m > 0, m < 0. $VBOEP VO FOUFSP m FT UBM RVF m > 0 EFDJNPT RVF EJDIP FOUFSP FT QPTJUJWP Z TJ TBUJTGBDF m < 0 EFDJNPT RVF FT OFHBUJWP JJ &O FM DBTP FO FM RVF m Z n TFBO OBUVSBMFT DPNP N ⊂ Z FOUPODFT MB SFMBDJÓO EF DPNQBSBDJÓO i < u SFDJÊO EFàOJEB FO Z DPJODJEF DPO MB EFàOJEB FO FM DPOKVOUP N FT EFDJS TJ EFOPUÃSBNPT FTUBT SFMBDJPOFT DPNP 0 4FBO d = (a, b) Z g = (ka, kb) 7FBNPT RVF kd = g. 5FOFNPT MPT TJHVJFOUFT IFDIPT  &T FWJEFOUF RVF d | a Z d | b. -VFHP dk | ak Z dk | bk MP RVF JNQMJDB kd ≤ g.  &T FWJEFOUF RVF g | ka Z g | kb. "EFNÃT FYJTUFO FOUFSPT x, y UBMFT RVF d = ax + by Z BTÎ kd = kax + kby &O DPOTFDVFODJB g | kd FOUPODFT g ≤ kd. %F MBT DPODMVTJPOFT FO  Z  TF PCUJFOF MB JHVBMEBE kd = g.

 

*OUSPEVDJNPT BIPSB PUSP UFNB JNQPSUBOUF %FàOJDJÓO  4FBO a Z b FOUFSPT OP OVMPT &M NÎOJNP DPNÙO NÙMUJQMP .$. EF a Z b EFOPUBEP QPS [a, b] TF EFàOF DPNP FM NFOPS FOUFSP QPTJUJWP RVF FT NÙMUJQMP DPNÙO EF a Z b &T EFDJS [a, b] = NÎO{m ∈ N∗ : a | m ∧ b | m}.



-PT OÙNFSPT FOUFSPT

0CTFSWBDJÓO &M DPOKVOUP EF NÙMUJQMPT QPTJUJWPT DPNVOFT FT OP WBDÎP QVFT ab Z −ab TPO NÙMUJQMPT DPNVOFT EF a Z b QPS UBOUP |ab| FT NÙMUJQMP DPNÙO EF a Z b -VFHP QPS FM 1#0 FYJTUF FM .$. EF a Z b &KFSDJDJP  1SVFCF RVF [a, b] = [−a, b] = [a, −b] = [−a, −b]. &M TJHVJFOUF UFPSFNB DBSBDUFSJ[B FM .$. 5FPSFNB  4FBO a Z b FOUFSPT OP OVMPT &OUPODFT m = [a, b] TJ Z TPMP TJ (i) m > 0

(ii) a | m ∧ b | m

(iii) a | n ∧ b | n ⇒ m | n.

%FNPTUSBDJÓO 1SPCFNPT MB DPOEJDJÓO OFDFTBSJB J Z JJ TF TJHVFO JO NFEJBUBNFOUF 7FBNPT RVF TF UJFOF JJJ  4VQPOHBNPT RVF n ∈ Z FT UBM RVF a | n ∧ b | n -VFHP n FT NÙMUJQMP DPNÙO EF a Z b Z QPS UBOUP m ≤ |n| 1PS FM BMHPSJUNP EF MB EJWJTJÓO |n| = mq + r QBSB BMHVOPT FOUFSPT q, r DPO 0 ≤ r < m. -B JEFB FT QSPCBS RVF r = 0. 4VQPOHBNPT FOUPODFT RVF r > 0 5FOJFOEP FO DVFOUB MB QSJNFSB BàSNBDJÓO FO JJ Z RVF a | n PCUFOFNPT RVF a | r %F MB NJTNB NBOFSB TF DPODMVZF RVF b | r Z QPS UBOUP r FT VO NÙMUJQMP DPNÙO EF a Z b Z DPNP m FT FM .$. TF UJFOF RVF m ≤ r MP RVF DPOUSBEJDF RVF r < m "TÎ RVFEB QSPCBEP RVF r = 0 Z FO DPOTFDVFODJB m | n. 7FBNPT BIPSB MB DPOEJDJÓO TVàDJFOUF TFB M = [a, b] Z EFNPTUSFNPT RVF M = m. 1PS EFàOJDJÓO a | M Z b | M  1PS JJJ

m | M Z BTÎ MMFHBNPT B RVF m ≤ M  1PS PUSP MBEP PCUFOFNPT EF JJ RVF M ≤ m MP DVBM OPT EB MB JHVBMEBE M = m. $PO  FTUP RVFEB EFNPTUSBEP FM UFPSFNB  5FPSFNB  4FBO a Z b FOUFSPT OP OVMPT &OUPODFT [a, b] · (a, b) = |ab|. %FNPTUSBDJÓO $PNP (a, b) = (a, −b) = (−a, b) = (−a, −b) Z [a, b] = [a, −b] = [−a, b] = [−a, −b] QPEFNPT TVQPOFS RVF a > 0 Z b > 0. 4FBO m = [a, b] Z d = (a, b). %F FTUP ÙMUJNP TBMF RVF d | a Z d | b Z QPS UBOUP FYJTUFO FOUFSPT ta Z tb UBMFT RVF a = dta Z b = dtb  1PS MB PCTFSWBDJÓO IFDIB FO MB QÃHJOB  UFOFNPT RVF (ta , tb ) = 1 $PNP ab = d2 ta tb QBSB QSPCBS FM UFPSFNB CBTUB WFS RVF m = dta tb . 1BSB FMMP WBNPT B BQMJDBS FM UFPSFNB BOUFSJPS ˆ > 0 7FBNPT JJ

FT EFDJS RVF EFOPUFNPT m ˆ = dta tb . &T DMBSP RVF J m m ˆ FT NÙMUJQMP DPNÙO EF a Z b Z JJJ

RVF m ˆ FT FM NFOPS EF MPT NÙMUJ QMPT DPNVOFT -B QSJNFSB QBSUF FT JONFEJBUB QVFT m ˆ = dta tb = atb Z



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

m ˆ = bta . 'JOBMNFOUF TVQPOHBNPT RVF a | n Z b | n WFBNPT RVF m ˆ | n &O FGFDUP FYJTUFO FOUFSPT r Z s UBMFT RVF n = ar = bs P FRVJWBMFOUF NFOUF rdta = sdtb EF MP DVBM TF JOàFSF RVF tb | rta Z DPNP (ta , tb ) = 1 FOUPODFT tb | r BTÎ QPEFNPT FTDSJCJS r = utb QBSB BMHÙO FOUFSP u -VFHP n = ar = autb = udta tb = um ˆ MP DVBM JNQMJDB RVF m ˆ | n %F FTUB  NBOFSB RVFEB EFNPTUSBEP FM UFPSFNB  $PSPMBSJP  4FBO a Z b FOUFSPT OP OVMPT &OUPODFT a Z b TPO QSJNPT SFMBUJWPT TJ Z TPMP TJ [a, b] = |ab|. $PSPMBSJP  4FBO a, b Z k FOUFSPT OP OVMPT &OUPODFT [ka, kb] = |k|[a, b]. %FNPTUSBDJÓO %F MBT JHVBMEBEFT TJHVJFOUFT TF EFEVDF FM DPSPMBSJP |k|[ka, kb](a, b) = [ka, kb](ka, kb) = k 2 |ab| = k 2 [a, b](a, b).   &KFSDJDJP  %FNVFTUSF FM TJHVJFOUF UFPSFNB TFBO a1 , a2 , . . . , an FO UFSPT OP OVMPT &OUPODFT d = (a1 , a2 , . . . , an ) TJ Z TPMP TJ J d > 0 JJ d | a1 , d | a2 , . . . , d | an JJJ g | a1 , g | a2 , . . . , g | an ⇒ g | d. &KFSDJDJP  %FNVFTUSF FM TJHVJFOUF UFPSFNB 4FBO a1 , a2 , . . . , an FO UFSPT OP OVMPT &OUPODFT m = [a1 , a2 , . . . , an ] TJ Z TPMP TJ J m > 0 JJ a1 | m, a2 | m, . . . , an | m JJJ a1 | k, a2 | k, . . . , an | k ⇒ m | k. 5FPSFNB  4FBO n ≥ 3 Z a1 , a2 , . . . , an FOUFSPT OP OVMPT &OUPODFT (a1 , a2 , . . . , an ) = ((a1 , a2 , . . . , an−1 ), an ). %FNPTUSBDJÓO 4FBO d = (a1 , a2 , . . . , an ), d = ((a1 , a2 , . . . , an−1 ), an ) Z d = (a1 , a2 , . . . , an−1 ) -VFHP d | ai QBSB UPEP i = 1, 2, . . . , n − 1 Z BTÎ d | d Z DPNP d | an FOUPODFT d | d  1PS PUSB QBSUF d | d Z d | ai QBSB UPEP i = 1, 2, . . . , n − 1 Z QVFTUP RVF d | an PCUFOFNPT RVF d | ai QBSB UPEP i = 1, 2, . . . , n. &O DPOTFDVFODJB d | d Z QPS UBOUP   d = d . &KFSDJDJP  4FBO n ≥ 3 Z a1 , a2 , . . . , an FOUFSPT OP OVMPT &OUPODFT [a1 , a2 , . . . , an ] = [[a1 , a2 , . . . , an−1 ], an ].



-PT OÙNFSPT FOUFSPT

&KFSDJDJP  4FBO a Z b FOUFSPT QPTJUJWPT UBMFT RVF [a, b] = (a, b). 1SVFCF RVF a = b. &KFNQMP  4FBO a, b Z c FOUFSPT 1SPCFNPT RVF MB FDVBDJÓO ax + by = c UJFOF TPMVDJPOFT FOUFSBT x, y TJ Z TPMP TJ (a, b) | c. -B OFDFTJEBE FT JONFEJBUB QVFTUP RVF (a, b) | a Z (a, b) | b FOUPODFT QPS MB IJQÓUFTJT (a, b) | c. 7FBNPT BIPSB MB DPOEJDJÓO TVàDJFOUF TFB g = (a, b). &OUPODFT FYJTUFO FOUFSPT x0 , y0 UBMFT RVF g = ax0 + by0  "EFNÃT DPNP g | c FOUPODFT c = gt QBSB BMHÙO FOUFSP t Z QPS UBOUP c = gt = a(x0 t) + b(y0 t) MP DVBM UFSNJOB MB EFNPTUSBDJÓO &KFSDJDJP  4FB p > 5 VO FOUFSP QSJNP 1SVFCF RVF OP FYJTUFO FOUFSPT x, y UBMFT RVF x + y = 200 DPO (x, y) = p. &KFSDJDJP  4FBO a, b Z d FOUFSPT UBMFT RVF (a, b) = 1 Z d | a 1SVFCF RVF (d, b) = 1. &KFSDJDJP  %FNVFTUSF RVF MB TVNB EF MPT DVBESBEPT EF EPT FOUFSPT JNQBSFT OP QVFEF TFS VO DVBESBEP QFSGFDUP &KFSDJDJP  1SVFCF RVF OP FYJTUF VO FOUFSP N UBM RVF N 2 + 2 FT NÙMUJQMP EF  &KFSDJDJP  3FTVFMWB MPT TJHVJFOUFT QSPCMFNBT B %FNVFTUSF RVF QBSB UPEP FOUFSP m TF DVNQMF RVF m5 − m FT NÙMUJQMP EF  C 1SVFCF RVF UPEP FOUFSP RVF FT DVBESBEP QFSGFDUP FT EFDJS DBEB VOP EF MPT FOUFSPT 0, 1, 4, 9, 16, · · · , n2 , · · · FT EF MB GPSNB 4m P CJFO 4m + 1 QBSB BMHÙO m ∈ Z. D %FNVFTUSF RVF BM NFOPT VOP EF m FOUFSPT DPOTFDVUJWPT FT EJWJTJCMF QPS m (m ≥ 1). E 4FBO a Z b FOUFSPT àKPT 1SVFCF RVF 3 | (a2 + b2 ) ⇐⇒ 3 | a ∧ 3 | b. F 4FB m ∈ Z 1SVFCF RVF (3m + 2, 5m + 3) = 1. G 4FBO p Z q OÙNFSPT OBUVSBMFT UBMFT p q = 40000 Z OJOHVOP EF FMMPT FT NÙMUJQMP EF  )BMMF MB TVNB p + q. &KFSDJDJP  4FBO a Z b FOUFSPT Z m ∈ Z, m > 0 %FDJNPT RVF a FT DPOHSVFOUF DPO b NÓEVMP m MP DVBM TF EFOPUB a ≡ b (NÓE m) TJ m | a − b. %FNVFTUSF



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

B -B SFMBDJÓO EF DPOHSVFODJB EFàOJEB FO Z FT VOB SFMBDJÓO EF FRVJ WBMFODJB -BT DMBTFT EF FRVJWBMFODJB HFOFSBEBT TF MMBNBO DMBTFT SFTJEVBMFT C 4J a Z b TPO DPOHSVFOUFT NÓEVMP m FOUPODFT a Z b EFKBO FM NJTNP SFTUP BM EJWJEJSMPT QPS m. D 4J a ≡ b (NÓE m) FOUPODFT m | a TJ Z TPMP TJ m | b. /PUB BMHVOPT FKFSDJDJPT QSFTFOUBEPT FO FTUF DBQÎUVMP GVFSPO FYUSBÎ EPT EFM UFYUP i 5FPSÎB EF OÙNFSPT QBSB QSJODJQJBOUFT u 

'BDVMUBE EF $JFODJBT 6OJWFSTJEBE /BDJPOBM EF $PMPNCJB EF MPT BVUPSFT - 3 +JNÊ OF[ #FDFSSB + & (PSEJMMP "SEJMB Z ( / 3VCJBOP 0SUFHÓO %F MB NJTNB NBOFSB BMHVOBT JEFBT Z EFNPTUSBDJPOFT QSFTFOUBEBT TF FYUSBDUBSPO EF BMMÎ PCWJBNFOUF DPO MB BVUPSJ[BDJÓO EF MPT BVUPSFT JBMBT`Qv2+iQ $POTVMUF TPCSF FM UFPSFNB GVOEBNFOUBM EF MB BSJU NÊUJDB FOVODJBEP QSVFCB BQMJDBDJPOFT FUD



$BQÎUVMP  -PT OÙNFSPT SBDJPOBMFT

5SBUBNPT FO FTUF DBQÎUVMP FM DPOKVOUP EF MPT OÙNFSPT SBDJPOBMFT EFTEF TV DPOTUSVDDJÓO IBTUB WFS RVF DPOTUJUVZFO VO DBNQP PSEFOBEP Z BS RVJNFEJBOP 6TBNPT DMBTFT EF FRVJWBMFODJB Z MB FTUSVDUVSB EF Z QBSB MB DPOTUSVDDJÓO EF EJDIPT OÙNFSPT &TUVEJBNPT UBNCJÊO MBT TVDFTJPOFT EF OÙNFSPT SBDJPOBMFT MP DVBM TFSÃ JNQPSUBOUF FO FM TJHVJFOUF DBQÎUVMP QBSB MB DPOTUSVDDJÓO EF MPT OÙNFSPT SFBMFT &O QSJODJQJP IBSFNPT VOB QSFTFOUBDJÓO EFM DPODFQUP EF DBNQP P DVFSQP

 &M DPODFQUP EF DBNQP &YJTUFO PUSBT FTUSVDUVSBT BMHFCSBJDBT DPNP BOJMMPT NÓEVMPT FUD RVF OP TPO PCKFUP EF FTUVEJP FO FTUF UFYUP Z RVF IBDFO QBSUF EF MBT BTJHOBUVSBT QPTUFSJPSFT EF ÃMHFCSB BCTUSBDUB 4JO FNCBSHP DPNP MPT OÙNFSPT SBDJP OBMFT Z MPT OÙNFSPT SFBMFT VOP EF MPT PCKFUJWPT QSJODJQBMFT EF FTUVEJP FO FTUF MJCSP DPOTUJUVZFO VO FKFNQMP QBSUJDVMBS EF FTUSVDUVSB BMHFCSBJDB MMBNBEB DBNQP JOUSPEVDJNPT FTUB OPDJÓO FO FTUF QVOUP "MHVOPT EFUB MMFT BEJDJPOBMFT TPCSF DBNQPT TF QSFTFOUBO FO FM BQÊOEJDF %FàOJDJÓO  6O DBNQP P VO DVFSQP FT VO DPOKVOUP F OP WBDÎP EPUBEP EF EPT PQFSBDJPOFT CJOBSJBT TVNB (+) Z QSPEVDUP (·) UBMFT RVF TF DVNQMFO MPT TJHVJFOUFT BYJPNBT MMBNBEPT BYJPNBT EF DBNQP P BMHFCSBJDPT 4" "TPDJBUJWJEBE EF + QBSB UPEP a, b, c ∈ F TF UJFOF RVF (a + b) + c = a + (b + c). 4/ &YJTUFODJB EF OFVUSP QBSB + (∃ 0 ∈ F) (∀ a ∈ F) (a + 0 = a) 4 * &YJTUFODJB EF JOWFSTPT QBSB + (∀ a ∈ F) (∃ (−a) ∈ F) (a + (−a) = 0).



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

4$ $PONVUBUJWJEBE EF + QBSB UPEP QBS a, b ∈ F TF DVNQMF RVF a + b = b + a 1" "TPDJBUJWJEBE EF i · u QBSB UPEP a, b, c ∈ F TF UJFOF RVF (a · b) · c = a · (b · c). 1/ &YJTUFODJB EF OFVUSP QBSB i · u (∃ 1 ∈ F  {0}) (∀ a ∈ F) (a · 1 = a). 1 * &YJTUFODJB EF JOWFSTPT QBSB i · u (∀ a ∈ F  {0}) (∃ (a−1 ) ∈ F) (a · a−1 = 1). 1$ $PONVUBUJWJEBE EF i · u QBSB UPEP QBS a, b ∈ F TF DVNQMF RVF a · b = b · a % %JTUSJCVUJWB QBSB UPEP a, b, c ∈ F TF DVNQMF RVF a · (b + c) = a · b + a · c. 0CTFSWBDJPOFT J -PT FMFNFOUPT 0 Z 1 TPO ÙOJDPT TVQPOHBNPT RVF FYJTUF 1 ∈ F UBM RVF QBSB UPEP x ∈ F, x · 1 = x &O QBSUJDVMBS QBSB x = 1 TF UJFOF RVF 1 · 1 = 1. 1PS PUSB QBSUF 1 · 1 = 1 Z QPS TFS i · u VOB PQFSBDJÓO CJOBSJB TF UJFOF RVF 1 = 1  -B VOJDJEBE EF 0 TF EFNVFTUSB EF NBOFSB TJNJMBS FKFSDJDJP  1PS UBOUP FTUPT FMFNFOUPT TF MMBNBO DFSP Z VOJEBE EF F SFTQFDUJWBNFOUF JJ -PT JOWFSTPT UBNCJÊO TPO ÙOJDPT WFS &KFSDJDJP   &M FMFNFOUP −a ∈ F TF MMBNB JOWFSTP BEJUJWP EF a Z FM FMFNFOUP a−1 ∈ F DPO a = 0 RVF TF EFOPUB UBNCJÊO QPS 1/a TF MMBNB SFDÎQSPDP P JOWFSTP NVMUJQMJDB UJWP EF a. JJJ /ÓUFTF RVF 1$ Z % JNQMJDBO MB EJTUSJCVUJWB B EFSFDIB FT EFDJS (b + c) · a = a · b + a · c. JW %F BIPSB FO BEFMBOUF TF PNJUJSÃ FM QVOUP FTUP FT TFHVJSFNPT FTDSJ CJFOEP ab FO MVHBS EF a · b 4F BDPTUVNCSB VTBS MB OPUBDJÓO  F, +, · QBSB FTQFDJàDBS RVF F FT VO DBNQP DPO MB TVNB i u Z FM QSPEVDUP i · u &O DBTP EF OP IBCFS BNCJHÛFEBE FTDSJCJSFNPT TJNQMFNFOUF F &KFSDJDJP  %FNVFTUSF MB VOJDJEBE EFM OFVUSP QBSB MB TVNB Z MB VOJ DJEBE EF MPT FMFNFOUPT JOWFSTPT FO VO DBNQP F



-PT OÙNFSPT SBDJPOBMFT

" DPOUJOVBDJÓO TF KVTUJàDBO NVDIBT EF MBT NBOJQVMBDJPOFT BMHFCSBJ DBT RVF BDPTUVNCSBNPT SFBMJ[BS FO VO DBNQP 4F EFNVFTUSBO BMHVOBT FM SFTUP DPOTUJUVZF VO FKFSDJDJP QBSB FM MFDUPS 5FPSFNB  $POTFDVFODJBT EF MPT BYJPNBT EF DBNQP  4FBO  F, +, · VO DBNQP Z a, b, c ∈ F &OUPODFT $ $BODFMBUJWB QBSB + TJ a + b = a + c FOUPODFT b = c $ a + b = 0 ⇒ a = −b $ a0 = 0 $ /P IBZ EJWJTPSFT EF DFSP FT EFDJS TJ ab = 0 FOUPODFT a = 0 Ó b = 0 $ a = −(−a) $ $BODFMBUJWB EFM QSPEVDUP TJ ab = ac Z a = 0 FOUPODFT b = c $ 4J ab = a Z a = 0 FOUPODFT b = 1 $ 4J ab = 1 Z a = 0 FOUPODFT b = a−1  $ 1BSB a = 0 (a−1 )−1 = a $ -FZ EF TJHOPT −(ab) = (−a)b = a(−b). $ -FZ EF TJHOPT (−a)(−b) = ab. %FNPTUSBDJÓO $ "M BQMJDBS MPT BYJPNBT 4/

4*

4"

4$ Z MB IJQÓUFTJT TF PCUJFOF b = b + 0 = b + [a + (−a)] = (b + a) + (−a) = (a + b) + (−a) = (a + c) + (−a) = (−a) + (a + c) = [−a + a] + c = 0 + c = c. $ a + b = 0 = −b + b $ Z 4$ JNQMJDBO a = −b $ 6TBOEP 4/ Z % TF UJFOF RVF a0 + 0 = a0 = a · (0 + 0) = a0 + a0 1PS $ TF DPODMVZF RVF a0 = 0 $ 4VQPOHBNPT a = 0 Z WFBNPT RVF b = 0. 1PS 1* FYJTUF a−1 ∈ F DPO a−1 = 0 Z UBM RVF aa−1 = 1. -VFHP VTBOEP MB IJQÓUFTJT 1$ Z 1"

UFOFNPT 0 = 0 · a−1 = (ab)a−1 = (ba)a−1 = b[aa−1 ] = b · 1 = b. $ 1PS $

DPNP a + (−a) = 0 TF UJFOF RVF a = −(−a) $ &M DÓNQVUP 0 = a0 = a[b + (−b)] = ab + a(−b) Z $ JNQMJDBO



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

a(−b) = −(ab) "TJNJTNP TF QSVFCB MB PUSB JHVBMEBE $ $BNCJBOEP a QPS −a FO MB TFHVOEB JHVBMEBE EF $ Z VTBOEP  $ TF UJFOF MP EFTFBEP  &KFSDJDJP  %FNVFTUSF MBT DPOTFDVFODJBT $

$

$ Z $ EFM 5FPSFNB  0CTFSWBDJÓO &T GÃDJM WFS RVF FO VO DBNQP F MB FDVBDJÓO a+x = b UJFOF QPS TPMVDJÓO ÙOJDB BM FMFNFOUP b + (−a) &TUP OPT QFSNJUF JOUSPEVDJS MB EFàOJDJÓO EF EJGFSFODJB FOUSF a Z b B TBCFS a − b := a + (−b) QBSB UPEP QBS a, b FO F %FàOJDJÓO  4FB F VO DBNQP %FDJNPT RVF F FT PSEFOBEP TJ FYJTUF VO TVCDPOKVOUP QSPQJP P EF F MMBNBEP EF MPT FMFNFOUPT QPTJUJWPT RVF TBUJTGBDF MPT TJHVJFOUFT BYJPNBT EF PSEFO •5SJDPUPNÎB QBSB DBEB a ∈ F FYBDUBNFOUF VOB EF MBT TJHVJFOUFT BàS NBDJPOFT TF DVNQMF a = 0,

a ∈ P,

−a ∈ P.

•$FSSBEVSB TJ a, b ∈ P FOUPODFT a + b ∈ P Z ab ∈ P  0CTFSWBDJÓO 4J F FT PSEFOBEP FM DPOKVOUP P EF FMFNFOUPT QPTJUJWPT RVFEB àKP /PUFNPT BEFNÃT RVF FM BYJPNB EF USJDPUPNÎB JNQMJDB RVF 0∈ / P  4J −a ∈ P EFDJNPT RVF a FT OFHBUJWP Z VTBNPT MB OPUBDJÓO −P QBSB EFTDSJCJS FM DPOKVOUP EF FMFNFOUPT OFHBUJWPT 'JOBMNFOUF OPUFNPT RVF F FT VOJÓO EJTKVOUB EF P, −P Z {0} *OUSPEVDJNPT BIPSB MB TJHVJFOUF OPUBDJÓO QBSB a ∈ F EFOPUBNPT a · a = a2

Z a0 := 1,

TJ

a = 0.

1SPQPTJDJÓO  4FBO F VO DBNQP PSEFOBEP Z a ∈ F, a = 0 &OUPODFT a2 ∈ P  &O QBSUJDVMBS 1 ∈ P  %FNPTUSBDJÓO $PNP a = 0 Z F FT PSEFOBEP FOUPODFT a ∈ P P CJFO −a ∈ P  &O FM QSJNFS DBTP FM BYJPNB EF DFSSBEVSB JNQMJDB RVF a2 ∈ P Z FO FM TFHVOEP FM NJTNP BYJPNB Z MB DPOTFDVFODJB $ HBSBOUJ[BO   RVF a2 ∈ P  'JOBMNFOUF DPNP 1 = 1 · 1 TF UJFOF RVF 1 ∈ P. &KFSDJDJP  1SVFCF RVF TJ MB FDVBDJÓO x2 + 1 = 0 UJFOF TPMVDJÓO FO FM DBNQP F FOUPODFT F OP QVFEF TFS PSEFOBEP



-PT OÙNFSPT SBDJPOBMFT

%FàOJDJÓO  4FBO F VO DBNQP PSEFOBEP Z P FM TVCDPOKVOUP EF FMFNFOUPT QPTJUJWPT 4F EFàOFO MBT TJHVJFOUFT SFMBDJPOFT QBSB a, b ∈ F B a < b :⇐⇒ b − a ∈ P  -ÊBTF a NFOPS RVF b C a > b :⇐⇒ b < a ⇐⇒ a − b ∈ P  -ÊBTF a NBZPS RVF b D a ≤ b :⇐⇒ ( a < b ∨ a = b ) ⇐⇒ ( b − a ∈ P -ÊBTF a NFOPS P JHVBM RVF b

∨

a = b )

E a ≥ b :⇐⇒ b ≤ a ⇐⇒ ( b < a ∨ a = b ) ⇐⇒ ( a − b ∈ P ∨ a = b ) -ÊBTF a NBZPS P JHVBM RVF b 0CTFSWBDJPOFT /PUFNPT RVF FO VO DBNQP PSEFOBEP J a > 0 ⇐⇒ a − 0 ∈ P ⇐⇒ a ∈ P  &TUP FT a > 0 TJ Z TPMP TJ a FT QPTJUJWP JJ a < 0 ⇐⇒ 0 − a ∈ P ⇐⇒ −a ∈ P  &TUP FT a < 0 TJ Z TPMP TJ a FT OFHBUJWP JJJ a > b TJ Z TPMP TJ a − b > 0 &TUP FT DJFSUP QPSRVF a > b TJ Z TPMP TJ a − b ∈ P TJ Z TPMP TJ a − b > 0 %F MB NJTNB NBOFSB TF EFEVDF a < b ⇐⇒ a − b < 0 -BT BOUFSJPSFT EFTJHVBMEBEFT WBMFO TJ MPT TÎNCPMPT < Z > TF TVTUJ UVZFO QPS ≤ Z ≥ SFTQFDUJWBNFOUF JW a < 0 ⇐⇒ 0 − a = −a ∈ P ⇐⇒ −a > 0 W &M BYJPNB EF USJDPUPNÎB TF QVFEF FOVODJBS BTÎ QBSB UPEP a ∈ F P CJFO a = 0 P CJFO a > 0 P CJFO a < 0 WJ &M BYJPNB EF USJDPUPNÎB JNQMJDB RVF a ≮ 0 ⇐⇒ a ≥ 0 Z a  0 ⇐⇒ a < 0. 1SPQPTJDJÓO  &O VO DBNQP PSEFOBEP F TF DVNQMF MP TJHVJFOUF B 4J QBSB UPEP e > 0 TF UJFOF RVF x < e FOUPODFT x ≤ 0 C 1BSB UPEP a ∈ F a < a + 1. D 4J a < b Z c > 0 FOUPODFT ac < bc E 4J a < b Z c < 0 FOUPODFT ac > bc &O QBSUJDVMBS a < b ⇐⇒ −a > −b.

F 4J a > 0 FOUPODFT a−1 > 0



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

G 4J a < 0 FOUPODFT a−1 < 0 H a = 0 TJ Z TPMP TJ a ≤ 0 Z a ≥ 0

I ab < 0 ⇐⇒ ( a > 0 ∧ b < 0 ) ∨ ( a < 0

∧

 b>0) 

J a < b ⇐⇒ a + c < b + c K -B SFMBDJÓO i < u FT DPNQBUJCMF DPO MB TVNB i + u EFàOJEB FO F FTUP FT a < b Z c < d JNQMJDB a + c < b + d L -B SFMBDJÓO i < u FT USBOTJUJWB a < b Z b < c JNQMJDB a < c %FNPTUSBDJÓO 4F EFNVFTUSBO BMHVOBT EF MBT BàSNBDJPOFT MBT EFNÃT DPOTUJUVZFO VO GÃDJM FKFSDJDJP B 4VQPOHBNPT RVF OP TF DVNQMF MB DPODMVTJÓO &OUPODFT QPS MB PCTFS WBDJÓO WJ QSFWJB x > 0 "QMJDBOEP MB IJQÓUFTJT DPO e = x TF PCUJFOF MB DPOUSBEJDDJÓO x < x -VFHP x ≤ 0 D /PUFNPT RVF QPS EFàOJDJÓO ac < bc TJ Z TPMP TJ bc − ac = (b − a)c ∈ P. "IPSB CJFO MB IJQÓUFTJT JNQMJDB b − a ∈ P Z c ∈ P  'JOBMNFOUF FM BYJPNB EF DFSSBEVSB HBSBOUJ[B RVF (b − a)c ∈ P  F &T FWJEFOUF RVF a−1 FYJTUF Z FT EJGFSFOUF EF DFSP 4J a−1 < 0 FOUPODFT QPS D TF UFOESÎB a−1 a < a · 0 Z MVFHP 1 < 0 $PNP DPOTFDVFODJB EF FTUB DPOUSBEJDDJÓO EFCF TFS a−1 > 0 H 4J a = 0 QPS USJDPUPNÎB MBT PUSBT EPT QPTJCJMJEBEFT OP TF DVNQMFO FT EFDJS a ≮ 0 Z a ≯ 0. 6TBOEP OVFWBNFOUF MB PCTFSWBDJÓO WJ TF DPODMVZF a ≥ 0 Z a ≤ 0 3FDÎQSPDBNFOUF TJ PDVSSF a ≥ 0 Z a ≤ 0 WJ JNQMJDB RVF a = 0 I 4J ab < 0 FOUPODFT a = 0 )BZ EPT DBTPT a > 0 P CJFO a < 0 4J TF UJFOF FM QSJNFSP VTBOEP F Z D TF DPODMVZF RVF b < 0 4J PDVSSF FM TFHVOEP VTBOEP G Z E TF UJFOF RVF b > 0 &TUP QSVFCB MB DPOEJDJÓO OFDFTBSJB 3FDÎQSPDBNFOUF DVBMRVJFSB EF MBT EPT TJUVBDJPOFT JNQMJDB RVF  −ab ∈ P FTUP FT ab < 0.  &KFSDJDJP  4FB F VO DVFSQP PSEFOBEP Z a, b ∈ F %FNVFTUSF B 0 < a < 1 ⇒ a2 < a C a > 1 ⇒ a2 > a D 4J a > 0 Z b > 0 FOUPODFT a < b ⇐⇒ a2 < b2 . E (a > 0 ∧ b > 0) ⇒ (a + b)−1 < a−1 .



-PT OÙNFSPT SBDJPOBMFT

F 1BSB UPEP a, b TF DVNQMF RVF a2 ± ab + b2 ≥ 0. G a2 + b2 = 0 ⇐⇒ a = b = 0 %FàOJDJÓO  4FBO F VO DBNQP PSEFOBEP Z A ⊂ F J %FDJNPT RVF A FT BDPUBEP TVQFSJPSNFOUF SFTQFDUJWBNFOUF BDPUBEP JOGFSJPSNFOUF TJ FYJTUF b ∈ F UBM RVF QBSB UPEP x ∈ A TF UJFOF x ≤ b SFTQFDUJWBNFOUF x ≥ b  VO FMFNFOUP b ∈ F DPO FTUB QSPQJFEBE TF MMB NB DPUB TVQFSJPS EF A DPUB JOGFSJPS EF A  A FT BDPUBEP TJ FT BDPUBEP TVQFSJPSNFOUF F JOGFSJPSNFOUF /ÓUFTF RVF TJ A = ∅ UPEP FMFNFOUP EF F FT DPUB TVQFSJPS SFTQFDUJWB NFOUF DPUB JOGFSJPS EF A &TUF DBTP OP PGSFDF JOUFSÊT FO FTUF MJCSP JJ &YUSFNP TVQFSJPS P TVQSFNP EF A FT VO FMFNFOUP s ∈ F RVF FT DPUB TVQFSJPS EF A Z UBM RVF TJ s ∈ F FT DVBMRVJFS DPUB TVQFSJPS EF A FOUPO DFT s ≤ s  FT EFDJS s FT MB NFOPS EF MBT DPUBT TVQFSJPSFT -P EFOPUBNPT QPS TVQ A "IPSB CJFO TJ TVQ A ∈ A TF MMBNB NÃYJNP EF A Z MP EFOP UBNPT QPS NÃY A JJJ &YUSFNP JOGFSJPS P ÎOàNP EF A FT VO FMFNFOUP y ∈ F RVF FT DPUB JOGFSJPS EF A UBM RVF TJ y  ∈ F FT DVBMRVJFS DPUB JOGFSJPS EF A FOUPODFT y  ≤ y FTUP FT y FT MB NBZPS EF MBT DPUBT JOGFSJPSFT &TDSJCJNPT ÎOG A Z FO FM DBTP EF RVF ÎOG A ∈ A MP MMBNBNPT NÎOJNP EF A Z TF SFQSFTFOUB QPS NÎO A 0CTÊSWFTF RVF VO DBNQP PSEFOBEP F OP UJFOF TVQSFNP FO F

QVFTUP RVF TJ s ∈ F MP GVFSB FOUPODFT s + 1 ≤ s MP DVBM DPOUSBEJDF C EF MB 1SPQPTJDJÓO  6O SB[POBNJFOUP TJNJMBS KVTUJàDB RVF F OP UJFOF ÎOàNP 0CTFSWBNPT UBNCJÊO RVF ∅ OP UJFOF TVQSFNP OJ ÎOàNP FO F &KFNQMP  4J F FT VO DBNQP PSEFOBEP FOUPODFT FM DPOKVOUP P EF FMFNFOUPT QPTJUJWPT FTUÃ BDPUBEP JOGFSJPSNFOUF QPS DFSP 5BNCJÊO FM DPO KVOUP −P EF FMFNFOUPT OFHBUJWPT FTUÃ BDPUBEP TVQFSJPSNFOUF QPS DFSP )BTUB BIPSB QBSB RVF VO DPOKVOUP F = ∅ EPUBEP EF EPT PQFSBDJPOFT CJOBSJBT TFB VO DBNQP EFCFO DVNQMJSTF MPT BYJPNBT BMHFCSBJDPT Z TJ BEFNÃT TF DVNQMFO MPT BYJPNBT EF PSEFO FM DBNQP FT PSEFOBEP )BZ VO UFSDFS UJQP EF BYJPNB MMBNBEP BYJPNB EF DPNQMFUF[ RVF DBSBDUFSJ[B MPT DBNQPT MMBNBEPT DPNQMFUPT %FàOJDJÓO  4FB F VO DBNQP PSEFOBEP %FDJNPT RVF F FT DPNQMFUP TJ UPEP TVCDPOKVOUP OP WBDÎP EF F BDPUBEP TVQFSJPSNFOUF UJFOF TVQSFNP FO F



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

" DPOUJOVBDJÓO TF EFNVFTUSB VO UFPSFNB RVF FYQSFTB PUSB GPSNB FRVJWBMFOUF QBSB FM BYJPNB EF DPNQMFUF[ &O MPT FKFSDJDJPT QSPQVFTUPT TF FODPOUSBSÃ PUSB FRVJWBMFODJB 5FPSFNB  4FB F VO DVFSQP PSEFOBEP &OUPODFT F FT DPNQMFUP TJ Z TPMP TJ UPEP TVCDPOKVOUP OP WBDÎP EF F BDPUBEP JOGFSJPSNFOUF UJFOF ÎOàNP FO F %FNPTUSBDJÓO 4VQPOHBNPT RVF F FT DPNQMFUP Z TFB A ⊂ F OP WBDÎP Z BDPUBEP JOGFSJPSNFOUF 7FBNPT RVF A UJFOF ÎOàNP FO F 1BSB FMMP TF DPOTUSVZF FM DPOKVOUP EF PQVFTUPT EF A FT EFDJS B = −A := {−x : x ∈ A}. $PNP A = ∅ FOUPODFT B = ∅ 1PS PUSB QBSUF FYJTUF y ∈ F DPUB JOGFSJPS EF A -VFHP TJ b ∈ B FOUPODFT −b ∈ A Z QPS UBOUP y ≤ −b Z BTÎ b ≤ −y &TUP QSVFCB RVF B FT BDPUBEP TVQFSJPSNFOUF QPS −y -B DPN QMFUF[ EF F HBSBOUJ[B RVF FYJTUF s ∈ F UBM RVF s = TVQ B 1SPCFNPT RVF −s FT FM ÎOàNP EF A &O FGFDUP TJ x ∈ A FOUPODFT −x ∈ B Z QPS FOEF −s ≤ x FTUP FT −s FT DPUB JOGFSJPS EF A "IPSB CJFO TJ y  FT DVBMRVJFS DPUB JOGFSJPS EF A FOUPODFT y  ≤ −b QBSB UPEP b ∈ B MP RVF RVJFSF EFDJS RVF −y  FT DPUB TVQFSJPS EF B $PNP s FT FM TVQSFNP EF B FOUPODFT s ≤ −y  . 1PS UBOUP y  ≤ −s MP DVBM HBSBOUJ[B RVF −s FT MB NBZPS EF MBT DPUBT JOGFSJPSFT EF A 2VFEB QSPCBEP RVF −s = ÎOG A 1BSB QSPCBS  FM SFDÎQSPDP TF QSPDFEF EF NBOFSB BOÃMPHB  0CTFSWBDJÓO %F MB EFNPTUSBDJÓO BOUFSJPS TF SFTBMUB FM TJHVJFOUF IFDIP TJ A = ∅ FT VO TVCDPOKVOUP UBM RVF ÎOG A FYJTUF FO VO DVFSQP PSEFOBEP F FOUPODFT TVQ(−A) FYJTUF Z BEFNÃT ÎOG A = − TVQ(−A). 1SPQPTJDJÓO  4J FM TVQSFNP Z FM ÎOàNP EF VO DPOKVOUP A FYJTUFO FOUPODFT TPO ÙOJDPT %FNPTUSBDJÓO 4VQPOHBNPT RVF TVQ A FYJTUF FO VO DBNQP F 4J s ∈ F FT UBNCJÊO TVQSFNP EF A FOUPODFT FO QBSUJDVMBS FT DPUB TVQFSJPS EF A Z QPS UBOUP s ≤ TVQ A %F MB NJTNB NBOFSB TVQ A ≤ s -VFHP s = TVQ A  1BSB MB VOJDJEBE EFM ÎOàNP CBTUB VTBS MB PCTFSWBDJÓO BOUFSJPS  5FPSFNB  $BSBDUFSJ[BDJÓO EFM TVQSFNP F ÎOàNP  4FBO F VO DBNQP PSEFOBEP ∅ = A ⊂ F BDPUBEP TVQFSJPSNFOUF FO F Z ∅ = B ⊂ F BDPUBEP JOGFSJPSNFOUF FO F &OUPODFT (i) s ∈ F FT DPUB TVQFSJPS EF A s = TVQ A ⇐⇒ 

(ii) (∀ e > 0) (∃ a ∈ A) (s − e < a)



-PT OÙNFSPT SBDJPOBMFT

y = ÎOG B ⇐⇒

(i) y ∈ F FT DPUB JOGFSJPS EF B (ii) (∀ e > 0) (∃ b ∈ B) (y + e > b).



%FNPTUSBDJÓO &T TVàDJFOUF QSPCBS 

ZB RVF VOB WF[ IFDIP FTUP MB BàSNBDJÓO  TF PCUJFOF GÃDJMNFOUF DPNP TF EFUBMMB B DPOUJOVBDJÓO y = ÎOG B ⇐⇒ y = − TVQ(−B) ⇐⇒ −y = TVQ(−B) ⇐⇒ ⇐⇒

(i) −y ∈ F FT DPUB TVQFSJPS EF −B (ii) (∀ e > 0) (∃ − b ∈ (−B)) (−y − e < −b) (i) y ∈ F FT DPUB JOGFSJPS EF B (ii) (∀ e > 0) (∃ b ∈ B) (y + e > b).

( ⇒ ) J TF EFTQSFOEF JONFEJBUBNFOUF EF MB EFàOJDJÓO EF TVQSFNP 1SPCFNPT JJ QPS FM BCTVSEP 4J FYJTUJFSB e > 0 UBM RVF QBSB UPEP a ∈ A TF DVNQMJFSB s − e ≥ a FOUPODFT s − e TFSÎB DPUB TVQFSJPS EF A Z QPS UBOUP s ≤ s − e $POUSBEJDDJÓO ( ⇐ ) 4VQPOHBNPT BIPSB RVF DPOUBNPT DPO FM MBEP EFSFDIP EF   7FBNPT RVF s FT MB NFOPS EF MBT DPUBT TVQFSJPSFT EF A 4VQPOHBNPT RVF FYJTUF s DPUB TVQFSJPS EF A DPO s < s &OUPODFT BQMJDBOEP MB IJQÓUFTJT DPO e = s − s > 0 FYJTUJSÎB a ∈ A DPO s < a MP DVBM FT BCTVSEP -VFHP   s ≤ s QBSB UPEB s DPUB TVQFSJPS EF A Z QPS UBOUP s = TVQ A. &KFSDJDJP  %FNPTUSBS RVF FM UFPSFNB BOUFSJPS TF QVFEF SFEBDUBS EF NBOFSB FRVJWBMFOUF BTÎ 4FBO F VO DBNQP PSEFOBEP ∅ = A ⊂ F BDPUBEP TVQFSJPSNFOUF FO F Z ∅ = B ⊂ F BDPUBEP JOGFSJPSNFOUF FO F &OUPODFT (i) s ∈ F FT DPUB TVQFSJPS EF A s = TVQ A ⇐⇒ (ii) (∀ x < s) (∃ a ∈ A) (x < a) (i) y = ÎOG B ⇐⇒ (ii)

y ∈ F FT DPUB JOGFSJPS EF B (∀ x > y) (∃ b ∈ B) (x > b).

'JOBMJ[BNPT FTUB TFDDJÓO SFTBMUBOEP PUSBT QSPQJFEBEFT EFM FYUSFNP TVQFSJPS Z EFM FYUSFNP JOGFSJPS



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

1SPQPTJDJÓO  4FBO A Z B TVCDPOKVOUPT EF VO DBNQP PSEFOBEP F &OUPODFT J 4J A Z B UJFOFO FYUSFNP TVQFSJPS FO F Z A ⊆ B FOUPODFT TVQ A ≤ TVQ B JJ 4J A Z B UJFOFO FYUSFNP JOGFSJPS FO F Z A ⊆ B FOUPODFT ÎOG A ≥ ÎOG B JJJ 4J A Z B UJFOFO FYUSFNP TVQFSJPS F JOGFSJPS FO F Z A ⊆ B FOUPODFT ÎOG B ≤ ÎOG A ≤ TVQ A ≤ TVQ B. %FNPTUSBDJÓO J 4FB x ∈ A 1PS MB IJQÓUFTJT SFTVMUB RVF x ∈ B $PNP TVQ B FYJTUF x ≤ TVQ B -VFHP TVQ B FT DPUB TVQFSJPS EF A Z QPS UBOUP TVQ A ≤ TVQ B JJ &T TJNJMBS B MB QSVFCB EF J   JJJ 1BSB x ∈ A TF UJFOF ÎOG A ≤ x ≤ TVQ A %F JJ Z J TF DPODMVZF  *OUSPEVDJNPT BIPSB MB TJHVJFOUF OPUBDJÓO QBSB a ∈ F Z n ∈ N∗ a1 := a Z QBSB n > 1, an := an−1 · a. 5BNCJÊO a−n := (a−1 )n Z a0 := 1 TJ a = 0. &KFSDJDJP  %FNVFTUSF RVF TJ F FT VO DBNQP Z a, b ∈ F FOUPODFT QBSB UPEP QBS EF OBUVSBMFT m ≥ 1 Z n ≥ 1 TF DVNQMF B an am = an+m  C (an )m = anm  D (ab)n = an bn  E am a−n = am−n Z (a−n )m = a−nm TJFNQSF RVF a = 0 "OBMJDF MB WBMJEF[ EF MBT QSPQJFEBEFT BOUFSJPSFT TJ m, n ∈ Z.

 &KFSDJDJPT TPCSF DBNQPT  1SVFCF RVF MB FDVBDJÓO cx = d DPO c = 0 UJFOF TPMVDJÓO ÙOJDB FO VO DBNQP F &TUP QFSNJUF JOUSPEVDJS MB OPUBDJÓO QBSB c = 0 d := d · c−1 . c  4FB F VO DBNQP Z TVQÓOHBTF RVF FYJTUF a ∈ F, a = 0 UBM RVF a + a = 0 1SVFCF RVF 1 + 1 = 0



-PT OÙNFSPT SBDJPOBMFT

 4FBO  F, +, · VO DBNQP Z ∅ = S ⊆ F %FNVFTUSF RVF  S, +, · FT VO DBNQP TVCDBNQP EF F TJ Z TPMP TJ (i) 1 ∈ S (ii) a, b ∈ S ⇒ a − b ∈ S (iii) a, b ∈ S ∧ b = 0 ⇒ a · b−1 ∈ S  4FBO F VO DVFSQP PSEFOBEP Z a ∈ F OP OVMP %FNVFTUSF a + a−1 ≥ 1 + 1,

P CJFO

a + a−1 ≤ −1 + (−1).

 4FBO F VO DBNQP Z a, b, c, d, α, β ∈ F DPO ad − bc = 0 3FTVFMWB FM TJTUFNB ax + by = α cx + dy = β, QBSB x Z QBSB y FO F.  4FB a ∈ F 6OB SBÎ[ DVBESBEB EF a FT VO FMFNFOUP b ∈ F UBM RVF b2 = a B y$VÃOUBT SBÎDFT DVBESBEBT UJFOF DFSP C 4VQÓOHBTF RVF a = 0. 1SVFCF RVF TJ 1 + 1 = 0 FOUPODFT a QPTFF B MP TVNP VOB SBÎ[ DVBESBEB Z RVF TJ 1 + 1 = 0 FOUPODFT FO DBTP EF RVF a UFOHB SBÎ[ DVBESBEB UJFOF FYBDUBNFOUF EPT  4FBO  F, +, · VO DVFSQP Z a ∈ F VO FMFNFOUP RVF DBSFDF EF SBÎ[ DVBESBEB FO F &TUF QSPCMFNB JOEJDB DPNP TF DPOTUSVZF VO DBNQP NÃT BNQMJP F RVF DPOUJFOF B F FO FM DVBM a UJFOF SBÎ[ DVBESBEB 4FB F = F × F EPUBEP EF MBT PQFSBDJPOFT  Z  EFàOJEBT QPS (x, y)(z, w) = (x+z, y+w), (x, y)(z, w) = (xz+ayw, xw+yz). J %FNVFTUSF RVF  F , ,   FT VO DBNQP JJ %FNPTUSBS RVF (x, 0)  (y, 0) = (x + y, 0) Z RVF (x, 0)  (y, 0) = (xy, 0). &TUP OPT QFSNJUF JEFOUJàDBS x DPO (x, 0) FT EFDJS TJ Ω := {(x, 0) : x ∈ F} Z TF EFàOF Φ : F → Ω QPS Φ(x) = (x, 0) FOUPODFT Φ FT VO JTPNPSàTNP &TUB BQMJDBDJÓO OPT QFSNJUF UBNCJÊO BàSNBS RVF F



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

DPOUJFOF B F FO FM TFOUJEP EF RVF F DPOUJFOF VO TVCDPOKVOUP Ω

FM DVBM FT JTPNPSGP B F JJJ )BMMF VOB SBÎ[ DVBESBEB EF (a, 0) FO F   "YJPNB EF DPOUJOVJEBE TFBO F VO DBNQP DPNQMFUP Z A, B ⊆ F OP WBDÎPT UBMFT RVF ∀ a ∈ A Z ∀ b ∈ B TF UJFOF RVF a ≤ b &OUPODFT FYJTUF c ∈ F UBM RVF ∀ a ∈ A Z ∀ b ∈ B, a ≤ c ≤ b %FNVFTUSF RVF FM BYJPNB EF DPOUJOVJEBE FT FRVJWBMFOUF BM BYJPNB EF DPNQMFUF[  4FB A VO TVCDPOKVOUP OP WBDÎP EF P DPO P FM DPOKVOUP EF FMF NFOUPT QPTJUJWPT EF VO DBNQP DPNQMFUP F 4J A FT BDPUBEP TVQF SJPSNFOUF QSVFCF RVF TVQ A > 0 &O DBTP EF RVF A TFB BDPUBEP JOGFSJPSNFOUF EFNVFTUSF RVF ÎOG A ≥ 0  4FBO F VO DBNQP DPNQMFUP Z A ⊂ F OP WBDÎP Z BDPUBEP 4J ÎOG A = TVQ A EFNVFTUSF RVF A FT VO DPOKVOUP TJOHVMBS FT EFDJS VOJUBSJP  &TUF FKFSDJDJP KVTUJàDB MB JEFOUJàDBDJÓO RVF TF BDPTUVNCSB SFBMJ[BS DVBOEP EPT FTUSVDUVSBT TPO JTPNPSGBT 4FBO F1 Z F2 EPT DBNQPT Z Φ : F1 → F2 VO JTPNPSàTNP FOUSF FTUPT DBNQPT FT EFDJS Φ FT CJZFDUJWP Z BEFNÃT Φ(a + b) = Φ(a) + Φ(b), Φ(ab) = Φ(a)Φ(b) %FNPTUSBS B Φ(0) = 0 C Φ(1) = 1 D Φ(−a) = −Φ(a)

−1 E 4J a = 0 FOUPODFT Φ(a−1 ) = Φ(a) 

F 4J MB FDVBDJÓO x2 +1 = 0 UJFOF TPMVDJÓO FO FM DBNQP F1 UBNCJÊO UJFOF TPMVDJÓO FO FM DBNQP F2  G 4J F1 FT PSEFOBEP UBNCJÊO MP FT F2  "EFNÃT a < b FO F1 JNQMJDB Φ(a) < Φ(b) FO F2  H 4J F1 FT DPNQMFUP UBNCJÊO MP FT F2   4FBO F VO DBNQP PSEFOBEP A Z B TVCDPOKVOUPT OP WBDÎPT EF F Z c ∈ F àKP 4F EFàOFO MPT TJHVJFOUFT DPOKVOUPT A + B := {a + b : a ∈ A ∧ b ∈ B}; AB := {ab : a ∈ A ∧ b ∈ B}; cA := {ca : a ∈ A}; c + A := {c + a : a ∈ A}.



-PT OÙNFSPT SBDJPOBMFT

%FNVFTUSF B 4J A Z B UJFOFO FYUSFNP TVQFSJPS FO F FOUPODFT A + B UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(A + B) = TVQ A + TVQ B C 4J c > 0 Z A UJFOF FYUSFNP TVQFSJPS FO F FOUPODFT cA UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(cA) = c · TVQ A D 4J A UJFOF FYUSFNP JOGFSJPS FO F Z c < 0 FOUPODFT cA UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(cA) = c· ÎOG A E 4J A Z B DPOUJFOFO TPMBNFOUF FMFNFOUPT QPTJUJWPT Z UJFOFO FYUSFNP TVQFSJPS FO F FOUPODFT AB UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(AB) = (TVQ A)(TVQ B) F 4J A Z B UJFOFO FYUSFNP TVQFSJPS FO F FOUPODFT A ∪ B UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(A∪B) = NÃY{TVQ A, TVQ B} G 4J A UJFOF FYUSFNP TVQFSJPS FO F FOUPODFT c + A UJFOF FYUSFNP TVQFSJPS FO F Z BEFNÃT TVQ(c + A) = c + TVQ A. H 4J A UJFOF FYUSFNP JOGFSJPS FO F FOUPODFT c + A UJFOF FYUSFNP JOGFSJPS FO F Z BEFNÃT ÎOG(c + A) = c+ ÎOG A.

 $POTUSVDDJÓO EF MPT OÙNFSPT SBDJPOBMFT %FàOJDJÓO  4FB B := Z × Z∗ := {(m, n) : m, n ∈ Z, n = 0}. %FàOJNPT FO B MB SFMBDJÓO , QPS (m, n)  (p, q) ⇐⇒ mq = np &T GÃDJM WFS RVF  FT VOB SFMBDJÓO EF FRVJWBMFODJB FO B %FOPUBSFNPT QPS [(m, n)] MB DMBTF EF FRVJWBMFODJB DVZP SFQSFTFOUBOUF FT (m, n) &KFSDJDJP  %FNVFTUSF RVF  FT VOB SFMBDJÓO EF FRVJWBMFODJB FO B %FàOJDJÓO  &O B TF EFàOFO MBT PQFSBDJPOFT  Z  EF MB TJHVJFOUF NBOFSB (m, n)  (p, q) := (mq + np, nq),

(m, n)  (p, q) := (mp, nq).

1SPQPTJDJÓO  -B SFMBDJÓO  EFàOJEB FO B FT DPNQBUJCMF DPO MBT PQFSBDJPOFT  Z  %FNPTUSBDJÓO 7FBNPT RVF  Z  TPO DPNQBUJCMFT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

4FBO (m, n), (m , n ), (p, q), (p , q  ) ∈ B UBMFT RVF (m, n)  (p, q) Z (m , n )  (p , q  ) -VFHP (m, n)  (m , n )  (p, q)  (p , q  ) ⇐⇒ (mn + nm , nn )  (pq  + qp , qq  ) ⇐⇒ qq  (mn + nm ) = nn (pq  + qp ) ⇐⇒ qq  mn + qq  nm = nn pq  + nn qp . "IPSB CJFO MB IJQÓUFTJT JNQMJDB RVF mq = np Z m q  = n p  .VM UJQMJDBOEP MB QSJNFSB EF FTUBT JHVBMEBEFT QPS q  n MB TFHVOEB QPS qn Z TVNBOEP FTUPT SFTVMUBEPT TF PCUJFOF MB JHVBMEBE RVF BQBSFDF FO MB ÙMUJNB FRVJWBMFODJB &TUP EFNVFTUSB MB DPNQBUJCJMJEBE EF  Z  6O  SB[POBNJFOUP TJNJMBS QSVFCB MB DPNQBUJCJMJEBE EF  Z  .  &KFSDJDJP  %FNVFTUSF RVF  Z  TPO DPNQBUJCMFT

 %FàOJDJÓO  &M DPOKVOUP EF MBT DMBTFT EF FRVJWBMFODJB B  MP MMBNBNPT FM DPOKVOUP EF MPT OÙNFSPT SBDJPOBMFT Z MP EFOPUBNPT QPS Q FT EFDJS Q := B  . %FDJS RVF q ∈ Q TJHOJàDB RVF q FT VOB DMBTF EF FRVJWBMFODJB FT EFDJS RVF FYJTUF (m, n) ∈ B UBM RVF q = [(m, n)] " DPOUJOVBDJÓO EPUBSFNPT B Q EF EPT PQFSBDJPOFT RVF TFHVJSFNPT TJN CPMJ[BOEP QPS i + u Z i · u QBSB OP IBDFS QFTBEB MB FTDSJUVSB Z QPS UBOUP FM DPOUFYUP JOEJDBSÃ TJ FM TÎNCPMP + SFTQFDUJWBNFOUF · FT MB TVNB MB NVMUJQMJDBDJÓO FO FM DPOKVOUP Z P FO FM DPOKVOUP Q -VFHP OPT PDVQB SFNPT EF QSPCBS RVF  Q, +, · FT VO DBNQP %FàOJDJÓO  %BEPT MPT SBDJPOBMFT [(m, n)] Z [(p, q)] EFàOJNPT MB TVNB Z MB NVMUJQMJDBDJÓO FOUSF FMMPT BTÎ [(m, n)]+[(p, q)] := [(m, n)(p, q)],

[(m, n)]·[(p, q)] := [(m, n)(p, q)].

-B DPNQBUJCJMJEBE EF  DPO MBT PQFSBDJPOFT  Z  OPT QFSNJUFO EF NPTUSBS RVF MB TVNB Z MB NVMUJQMJDBDJÓO EFàOJEBT FO Q TPO PQFSBDJPOFT CJOBSJBT Z NÃT BÙO OP EFQFOEFO EFM SFQSFTFOUBOUF EF DMBTF FTDPHJEP &O FGFDUP TJ [(m, n)] = [(m , n )] Z [(p, q)] = [(p , q  )] FTUP FT (m, n)  (m , n )

Z (p, q)  (p , q  ),

QPS MB DPNQBUJCJMJEBE EF  Z  PCUFOFNPT RVF (m, n)  (p, q)  (m , n )  (p , q  )



-PT OÙNFSPT SBDJPOBMFT

Z FTUP RVJFSF EFDJS RVF [(m, n)  (p, q)] = [(m , n )  (p , q  )] 1PS UBOUP [(m, n)] + [(p, q)] = [(m , n )] + [(p , q  )]. -VFHP MB TVNB i + u FT VOB PQFSBDJÓO CJOBSJB FO Q %F NBOFSB TJNJMBS TF WF RVF i · u FT VOB PQFSBDJÓO CJOBSJB FO Q 5FPSFNB   Q, +, · FT VO DBNQP %FNPTUSBDJÓO -B QSVFCB EF RVF TF DVNQMFO MPT BYJPNBT 4"

4$

1"

1$ Z % FT NVZ TJNQMF QVFT CBTUB VTBS MB EFàOJDJÓO EF i + u Z EF i · u EFàOJEBT FO Q Z MBT SFTQFDUJWBT QSPQJFEBEFT FO Z 7FBNPT MPT PUSPT BYJPNBT EF DBNQP 4/

4*

1/ Z 1*  TFB [(m, n)] ∈ Q /ÓUFTF RVF [(m, n)] + [(0, 1)] = [(m, n)]. &TUP EFNVFTUSB RVF [(0, 1)] FT OFVUSP EF MB TVNB 5BNCJÊO TF UJFOF RVF [(m, n)] + [(−m, n)] = [(0, n2 )] = [(0, 1)], EPOEF MB ÙMUJNB JHVBMEBE TF KVTUJàDB QPSRVF (0, n2 )  (0, 1) -VFHP DBEB OÙNFSP SBDJPOBM [(m, n)] QPTFF VO PQVFTUP BEJUJWP FO Q B TBCFS [(−m, n)] &M FMFNFOUP [(1, 1)] ∈ Q FT OFVUSP QBSB FM QSPEVDUP QVFT [(m, n)] · [(1, 1)] = [(m, n)] 'JOBMNFOUF EBEP [(m, n)] ∈ Q  {[(0, 1)]} FM SBDJPOBM [(n, m)] FT UBM RVF [(m, n)] · [(n, m)] = [(1, 1)] 1PS UBOUP TF  DVNQMF FM BYJPNB 1*  %F FTUB NBOFSB RVFEB EFNPTUSBEP FM UFPSFNB  0CTFSWBDJÓO -BT DPOTFDVFODJBT $  $ FOVODJBEBT FO FM 5FPSF NB  TPO WÃMJEBT FO DVBMRVJFS DBNQP F Z FO QBSUJDVMBS FO FM DBNQP  Q, +, · *HVBM WBMJEF[ UJFOFO MBT PCTFSWBDJPOFT IFDIBT EFTQVÊT EF MB EFàOJDJÓO EF DBNQP 1PS FMMP MPT FMFNFOUPT [(0, 1)] Z [(1, 1)] FOVODJBEPT FO MB EFNPTUSBDJÓO BOUFSJPS MPT MMBNBNPT FM DFSP Z FM VOP SFTQFDUJWB NFOUF EF  Q, +, · &T CVFOP FOGBUJ[BS UBNCJÊO RVF QBSB [(m, n)] ∈ Q FODPOUSBNPT RVF TV PQVFTUP BEJUJWP FT [(−m, n)] QFSP MB OPUBDJÓO DMÃ TJDB RVF TF IB WFOJEP VTBOEP FO VO DBNQP F QBSB EFOPUBS FM PQVFTUP BEJUJWP EF VO FMFNFOUP a ∈ F IB TJEP −a -VFHP QPEFNPT FTDSJCJS −[(m, n)] := [(−m, n)]. 'JOBMNFOUF TJ [(m, n)] ∈ Q  {[(0, 1)]}, TV SFDÎQSPDP RVF EFOPUBNPT QPS [(m, n)]−1 FT [(n, m)] Z FTUP TJHOJàDB RVF [(m, n)]−1 = [(n, m)].



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

" DPOUJOVBDJÓO WFSFNPT PUSPT IFDIPT JNQPSUBOUFT TPCSF Q &M QSJNFSP EF FMMPT FT RVF Q ⊃ Z NÃT FYBDUBNFOUF Q DPOUJFOF VO TVCDPOKVOUP QSPQJP RVF FT JTPNPSGP B Z 1BSB WFS FTUP EFàOBNPT FM DPOKVOUP    := [(m, 1)] : m ∈ Z Z  QPS j(m) := [(m, 1)] &TUB BQMJDBDJÓO SFTVMUB Z MB BQMJDBDJÓO j : Z → Z TFS VO JTPNPSàTNP FO FGFDUP TJ j(m) = j(n) FOUPODFT [(m, 1)] = [(n, 1)] ⇐⇒ (m, 1)  (n, 1) ⇐⇒ m = n, FT EFDJS j FT VOP B VOP -B TPCSFZFDUJWJEBE FT DMBSB 1PS PUSB QBSUF j(m + n) = [(m + n, 1)] = [(m, 1)  (n, 1)] = [(m, 1)] + [(n, 1)] = j(m) + j(n). %F NBOFSB TJNJMBS TF WF RVF j(mn) = j(m)j(n)  ⊂ Q FTUP FT FM FOUFSP &TUF JTPNPSàTNP OPT QFSNJUF JEFOUJàDBS Z DPO Z  TPMP TF EJGFSFODJBO m MP JEFOUJàDBNPT DPO FM SBDJPOBM [(m, 1)] ∈ Z FO MB FTDSJUVSB Z FTUP OPT GBDVMUB QBSB BCVTBS EF MB OPUBDJÓO Z FTDSJCJS [(m, 1)] = m "IPSB CJFO DPO FTUB JEFOUJàDBDJÓO FO NFOUF EBEP m ∈ Z 1 OP OVMP UJFOF TFOUJEP IBCMBS FO Q EF TV SFDÎQSPDP m−1 = m

FM DVBM −1 FT VO FMFNFOUP EF Q RVF JEFOUJàDBSFNPT DPO [(m, 1)] Z BCVTBOEP OVFWBNFOUF EF MB OPUBDJÓO FTDSJCJNPT [(m, 1)]−1 = m−1 . $PO UPEP FTUP MMFHBNPT B MP TJHVJFOUF QBSB DVBMRVJFS SBDJPOBM [(m, n)] TF UJFOF [(m, n)] = [(m, 1)  (1, n)] = [(m, 1)] · [(1, n)] = [(m, 1)] · [(n, 1)]−1 = m · n−1 , EPOEF MB ÙMUJNB JHVBMEBE FT SFTVMUBEP EF MB JEFOUJàDBDJÓO EF [(m, 1)] Z [(n, 1)]−1 DPO m Z n−1 SFTQFDUJWBNFOUF &O DPODMVTJÓO VO OÙNFSP Z EF BIPSB FO SBDJPOBM [(m, n)] MP JEFOUJàDBNPT DPO MB GSBDDJÓO m n BEFMBOUF MP QPEFNPT TFHVJS FTDSJCJFOEP EF FTUB GPSNB P UBNCJÊO DPNP m · n−1 DPO n = 0 &O QBSUJDVMBS TJ m ∈ Z, m = [(m, 1)] = m 1 . &O PUSPT UÊSNJOPT FM DPOKVOUP EF OÙNFSPT SBDJPOBMFT RVFEB EFTDSJUP EF MB TJHVJFOUF NBOFSB   Q= m : m, n ∈ Z, n =  0 . n -B TJHVJFOUF QSPQPTJDJÓO FYQSFTB RVF VO OÙNFSP SBDJPOBM TF QVFEF TV QPOFS TJFNQSF DPO EFOPNJOBEPS QPTJUJWP



-PT OÙNFSPT SBDJPOBMFT

1SPQPTJDJÓO  %BEP [(m, n)] ∈ Q FYJTUF [(p, q)] ∈ Q DPO q > 0 UBM RVF [(m, n)] = [(p, q)]. %FNPTUSBDJÓO $PNP n FT VO FOUFSP EJTUJOUP EF DFSP MB QSPQJFEBE EF USJDPUPNÎB FO Z JNQMJDB RVF n > 0 P CJFO n < 0 &O FM QSJNFS DBTP UPNBNPT p = m Z q = n 4J n < 0 TF FTDPHF p = −m Z q = −n Z FT  DMBSP RVF [(m, n)] = [(−m, −n)].  /PT PDVQBSFNPT BIPSB EF WFS RVF Q FT VO DBNQP PSEFOBEP p %FàOJDJÓO  4FBO m n Z q OÙNFSPT SBDJPOBMFT DPO n Z q FOUFSPT QPTJUJWPT %FàOJNPT MBT TJHVJFOUFT SFMBDJPOFT EF DPNQBSBDJÓO p m < :⇐⇒ mq < np EPOEF MB SFMBDJÓO mq < np FT MB EF Z J

n q p p m m > :⇐⇒ < ⇐⇒ np < mq JJ

n q q n m p p m p m ≤ :⇐⇒ < ∨ = ; JJJ

n q n q n q p p m m ≥ :⇐⇒ ≤ . JW

n q q n

7BMF BOPUBS RVF DVBOEP MPT OÙNFSPT SBDJPOBMFT B DPNQBSBS TPO FO UFSPT MB SFMBDJÓO < EFàOJEB FO Q DPJODJEF DPO MB EF Z &O FGFDUP TJ m, n ∈ Z Z TJ EFOPUBNPT NPNFOUÃOFBNFOUF QPS 0, WFNPT RVF QBSB UPEP n ∈ N, |rn | ≤ M. 1PS UBOUP MB TVDFTJÓO {rn } FT  BDPUBEB  &KFSDJDJP  4FB {qn } VOB TVDFTJÓO EF SBDJPOBMFT RVF DPOWFSHF B q %FNVFTUSF VTBOEP BSHVNFOUPT ε−N, RVF |qn | → |q| Z qn2 → q 2 . "OBMJDF MPT SFDÎQSPDPT %FàOJDJÓO  6OB TVDFTJÓO EF SBDJPOBMFT DPOWFSHFOUF DPO MÎNJUF DFSP FT MMBNBEB TVDFTJÓO OVMB &O MB TJHVJFOUF QSPQPTJDJÓO SFTVNJNPT BMHVOBT QSPQJFEBEFT EF MBT TV DFTJPOFT OVMBT 1SPQPTJDJÓO  4FBO {qn } Z {rn } TVDFTJPOFT EF SBDJPOBMFT B 4J {qn } FT OVMB FOUPODFT {−qn } UBNCJÊO FT OVMB C 4J {qn } Z {rn } TPO OVMBT FOUPODFT {qn + rn } FT OVMB D 4J {qn } FT BDPUBEB Z {rn } FT OVMB FOUPODFT {qn rn } FT OVMB &O QBSUJDVMBS FM QSPEVDUP EF EPT TVDFTJPOFT OVMBT UBNCJÊO FT OVMB %FNPTUSBDJÓO B &T JONFEJBUP QVFTUP RVF | − qn | = |qn |. C 4FB ε ∈ Q+  &YJTUF VO OBUVSBM N UBM RVF  ε ε ∧ |rn | < . n ≥ N ⇒ |qn | < 2 2 &O SFBMJEBE FTUF N TF PCUJFOF DPNP VO NÃYJNP FOUSF PUSPT EPT OBUVSBMFT N1 Z N2 UBM DVBM TF IJ[P FO MB QSVFCB EFM 5FPSFNB   -B EFTJHVBMEBE USJBOHVMBS JNQMJDB RVF TJ n ≥ N FOUPODFT |qn +rn | < ε. 1PS UBOUP RVFEB QSPCBEB MB BàSNBDJÓO D 4FB M ∈ Q+ UBM RVF QBSB UPEP n ∈ N, |qn | ≤ M  4FB ε ∈ Q+  $PNP rn → 0 FYJTUF N ∈ N UBM RVF TJ n ≥ N FOUPODFT |rn | < ε/M. -VFHP TJ   n ≥ N TF UJFOF RVF |qn rn | < ε Z FTUP QSVFCB RVF {qn rn } FT OVMB



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP  -B TVDFTJÓO n ≥ 1

EPOEF

∞ 1 n n=1

n2 n3 + 2

∞ FT OVMB QVFTUP RVF QBSB DBEB n=1

n2 1 n3 = · , n3 + 2 n n3 + 2    n3   < 1 FT OVMB Z QBSB DBEB n ≥ 1  3 n + 2

0CTFSWBDJÓO &T JONFEJBUP WFS RVF VOB TVDFTJÓO {rn } EF SBDJPOBMFT DPOWFSHF BM SBDJPOBM r TJ Z TPMP TJ MB TVDFTJÓO {rn − r} FT VOB TVDFTJÓO OVMB $PO FTUP Z DPO MB QSPQPTJDJÓO BOUFSJPS TF EFEVDFO MBT TJHVJFOUFT QSPQJFEBEFT  4J {rn } DPOWFSHF B r Z {qn } DPOWFSHF B q FOUPODFT {rn +qn } Z {rn qn } TPO DPOWFSHFOUFT Z UJFOFO QPS MÎNJUF B r + q Z rq SFTQFDUJWBNFOUF -B TFHVOEB BàSNBDJÓO QPS FKFNQMP TF EFEVDF EF MB EFTJHVBMEBE |rn qn − rq| ≤ |rn ||qn − q| + |q||rn − r| Z EFM IFDIP EF RVF {rn } FT BDPUBEB  4J q = 0 FOUPODFT 1/qn → 1/q QSJNFSP WFBNPT RVF B QBSUJS EF DJFSUP ÎOEJDF MPT UÊSNJOPT EF MB TVDFTJÓO TPO EJGFSFOUFT EF DFSP Z NÃT BÙO FYJTUF VOB DPOTUBOUF QPTJUJWB C UBM RVF |qn | > C QBSB UPEP n B QBSUJS EF EJDIP ÎOEJDF $PNP |q| > 0 MB EFàOJDJÓO EF DPOWFSHFODJB OPT |q|

UBM RVF QFSNJUF IBMMBS VO OBUVSBM N DPSSFTQPOEJFOUF B ε = 2 n ≥ N ⇒ |q − qn | <

|q| . 2

%F FTUP TF TJHVF RVF |q| − |qn | ≤ |q − qn | < |q| 2 TJFNQSF RVF n ≥ N Z QPS UBOUP |qn | > |q|/2 := C QBSB UPEP n ≥ N FT EFDJS MPT UÊSNJOPT EF MB TVDFTJÓO TPO EJGFSFOUFT EF DFSP QPS MP NFOPT B QBSUJS EFM UÊSNJOP EF MVHBS N  4FB ε ∈ Q+  /VFWBNFOUF MB DPOWFSHFODJB OPT QSPQPSDJPOB VO OBUVSBM N  UBM RVF TJ n ≥ N  FOUPODFT 1 |qn − q| < q 2 ε. 2 -VFHP TJ n ≥ NÃY{N, N  } TF UJFOF RVF   1   − 1  = |q − qn | < ε.  qn q  |q||qn | 

-PT OÙNFSPT SBDJPOBMFT



rn qn



1BSUJDVMBSNFOUF TF EFEVDF RVF TJ q = 0 FOUPODFT MB TVDFTJÓO r DPOWFSHF Z UJFOF QPS MÎNJUF BM SBDJPOBM · q 0CTFSWFNPT UBNCJÊO RVF DPO FTUBT QSPQJFEBEFT Z UFOJFOEP FO DVFO UB RVF 1/n → 0, 1/n2 → 0 Z 1/n3 → 0 DVBOEP n → ∞, FKFSDJDJPT DPNP FM JMVTUSBEP FO FM &KFNQMP  TF QVFEFO SFTPMWFS EF GPSNB NÃT TFODJMMB QPS FKFNQMP MÎNn→∞

4n3 − 7 4 − 7/n3 4 = MÎN = . n→∞ 3 2 3 3n − 9n − 5 3 − 9/n − 5/n 3

&KFSDJDJP  1SVFCF FM 5FPSFNB EFM TBOEXJDI QBSB TVDFTJPOFT FO Q TFBO {rn }, {sn } Z {qn } TVDFTJPOFT EF OÙNFSPT SBDJPOBMFT UBMFT RVF FYJTUF N0 DPO rn ≤ sn ≤ qn ∀n ≥ N0 . 4J rn → L ∈ Q Z qn → L, FOUPODFT sn → L. &KFSDJDJP  %FNVFTUSF RVF MBT TJHVJFOUFT TVDFTJPOFT TPO OVMBT (−1)n n n2 + 2

B

pn :=

C

r0 = 0

D

1 qn := n!

Z TJ n ≥ 1 rn :=

n  i=0

1 (n + i)2

"ZVEB VTF FM 5FPSFNB EFM TBOEXJDI QBSB C Z D  " DPOUJOVBDJÓO TF JOUSPEVDF VOB DMBTF FTQFDJBM EF TVDFTJÓO &TUB TF EJTUJOHVF EF MBT EFNÃT TVDFTJPOFT QPSRVF B QBSUJS EF DJFSUP ÎOEJDF FO BEFMBOUF MB EJGFSFODJB FOUSF EPT EF TVT UÊSNJOPT FT UBO QFRVFÒB DPNP VOP EFTFF %FàOJDJÓO  4FB {rn } VOB TVDFTJÓO EF OÙNFSPT SBDJPOBMFT %FDJNPT RVF {rn } FT VOB TVDFTJÓO EF $BVDIZ TJ QBSB DBEB ε ∈ Q+ FT QPTJCMF FODPOUSBS VO OBUVSBM N RVF QVFEF EFQFOEFS EF ε

UBM RVF QBSB DBEB QBS EF OBUVSBMFT m Z n DPO n > N Z m > N TF UJFOF RVF |rn − rm | < ε. &O TÎNCPMPT MB EFàOJDJÓO TF FTDSJCF {rn } FT EF $BVDIZ TJ (∀ ε ∈ Q+ ) (∃ N ∈ N) (∀ n)(∀ m) (n > N ∧ m > N ⇒ |rn −rm | < ε). 4F EFEVDF FOUPODFT RVF VOB TVDFTJÓO {rn } OP FT EF $BVDIZ TJ (∃ ε0 ∈ Q+ ) (∀N ∈ N) (∃ n)(∃ m) (n > N ∧ m > N ∧ |rn −rm | ≥ ε0 ), FT EFDJS FYJTUF VOB DPOTUBOUF SBDJPOBM QPTJUJWB ε0 UBM RVF EBEP DVBMRVJFS OBUVSBM N FT QPTJCMF IBMMBS PUSPT OBUVSBMFT n Z m NBZPSFT RVF N DPO



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

MB QSPQJFEBE |rn − rm | ≥ ε0 EBEP VO OBUVSBM N FYJTUFO EPT UÊSNJOPT EF MB TVDFTJÓO DPO ÎOEJDFT NBZPSFT RVF N TVàDJFOUFNFOUF BMFKBEPT  

2n FT EF $BVDIZ &M TJHVJFOUF &KFNQMP  -B TVDFTJÓO {rn } = n+1 DÃMDVMP OPT EJDF DÓNP DPOTUSVJS FM BSHVNFOUP QBSB DVBMRVJFS QBS EF OBUVSBMFT n Z m TF UJFOF RVF    2n 2|n − m| 2m  2(n + m)   n + 1 − m + 1  = (n + 1)(m + 1) ≤ (n + 1)(m + 1) 2(n + m + 2) < (n + 1)(m + 1)   n+1 m+1 =2 + (n + 1)(m + 1) (n + 1)(m + 1) 2 2 + · = m+1 n+1 "IPSB CJFO EBEP DVBMRVJFS ε ∈ Q+ MB QSPQJFEBE BSRVJNFEJBOB OPT 1 ε QSPWFF VO OBUVSBM N ≥ 1 UBM RVF < Z FO DPOTFDVFODJB TJ n > N N 4 Z m > N FOUPODFT 2 2 2 2 2 2 4 + < + < + = < ε. m+1 n+1 m n N N N 3FTVNJFOEP IFNPT QSPCBEP RVF EBEP ε ∈ Q+ FYJTUF N ∈ N DPOTFHVJEP QPS MB BSRVJNFEJBOB UBM RVF TJ n > N Z m > N FOUPODFT |rn −rm | < ε MP DVBM QSVFCB RVF EJDIB TVDFTJÓO FT EF $BVDIZ &KFSDJDJP  6TBOEP MB EFàOJDJÓO EFNVFTUSF RVF MBT TVDFTJPOFT

2  n  1 n −1 , TPO EF $BVDIZ TVHFSFODJB } EPOEF q := Z {q n n 2 n +1 k! ∀ k ∈ N TF WFSJàDB RVF k! ≥ 2k−1 

k=0

&KFSDJDJP  4FB {rn } VOB TVDFTJÓO EF $BVDIZ DPO rn ∈ Z QBSB UPEP n 1SVFCF RVF {rn } FT DPOTUBOUF FO ÙMUJNB JOTUBODJB FT EFDJS FYJTUFO DPOTUBOUFT C ∈ Z Z VO OBUVSBM N UBMFT RVF rn = C QBSB UPEP n ≥ N. &YJTUF VO HSBO OÙNFSP EF TVDFTJPOFT EF $BVDIZ DPNP TF KVTUJàDB B DPOUJOVBDJÓO 5FPSFNB  5PEB TVDFTJÓO EF SBDJPOBMFT DPOWFSHFOUF FT VOB TVDFTJÓO EF $BVDIZ



-PT OÙNFSPT SBDJPOBMFT

%FNPTUSBDJÓO 4FB {rn } VOB TVDFTJÓO EF SBDJPOBMFT DPOWFSHFOUF DPO MÎNJUF r ∈ Q Z TFB ε ∈ Q+  1PS MB DPOWFSHFODJB FYJTUF N ∈ N UBM RVF TJ n ≥ N FOUPODFT |rn − r| < 2ε  -VFHP TJ n > N Z m > N FOUPODFT |rn − rm | ≤ |rn − r| + |r − rm | <

ε ε + = ε. 2 2

&TUP EFNVFTUSB FM UFPSFNB

 

6O QBS EF UFPSFNBT NÃT TFSÃO QSFTFOUBEPT BOUFT EF FOUSBS FO MPT EFUBMMFT EF MB DPOTUSVDDJÓO EFM DPOKVOUP EF MPT OÙNFSPT SFBMFT 5FPSFNB  5PEB TVDFTJÓO EF $BVDIZ EF OÙNFSPT SBDJPOBMFT FT BDPUBEB %FNPTUSBDJÓO 4FB {rn } VOB TVDFTJÓO EF $BVDIZ EF OÙNFSPT SBDJPOBMFT 1BSB ε = 1 FYJTUF VO OBUVSBM N UBM RVF TJ n > N Z m > N FOUPODFT |rm − rn | < 1 &O QBSUJDVMBS àKBOEP VOB EF MBT WBSJBCMFT QPS FKFNQMP m = N + 1 Z VTBOEP MB EFTJHVBMEBE USJBOHVMBS PCUFOFNPT RVF QBSB DBEB n > N, |rn | < 1 + |rN +1 | := M  1BSB DPOUSPMBS UPEPT MPT UÊSNJOPT rn CBTUB UPNBS C := NÃY{|r0 |, |r1 |, . . . , |rN |, M } Z BTÎ TF DPODMVZF RVF   |rn | ≤ C QBSB UPEP n ∈ N. 5FPSFNB  4FBO {rn } Z {qn } TVDFTJPOFT EF $BVDIZ FO Q &OUPODFT {rn + qn } Z {rn qn } TPO TVDFTJPOFT EF $BVDIZ FO Q %FNPTUSBDJÓO 4FB ε ∈ Q+  $PNP {rn } Z {qn } TPO EF $BVDIZ FYJTUFO OBUVSBMFT N1 Z N2 UBMFT RVF n > N1 ∧ m > N1 ⇒ |rn − rm | < ε/2 n > N2 ∧ m > N2 ⇒ |qn − qm | < ε/2. -VFHP TJ n, m > NÃY{N1 , N2 } FOUPODFT |(rn + qn ) − (rm + qm )| ≤ |rn − rm | + |qn − qm | < ε. &TUP EFNVFTUSB MB QSJNFSB QBSUF EFM UFPSFNB 1PS FM UFPSFNB BOUFSJPS FYJTUF VOB DPOTUBOUF M ∈ Q+ UBM RVF |rn | ≤ M Z |qn | ≤ M QBSB UPEP n 3B[POBOEP DPNP BOUFT QPEFNPT FODPOUSBS VO OBUVSBM N  UBM RVF n > N  ∧ m > N  ⇒ |rn − rm | < ε/2M n > N  ∧ m > N  ⇒ |qn − qm | < ε/2M, Z FO DPOTFDVFODJB



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

|rn qn − rm qm | ≤ |rn qn − rn qm | + |rn qm − rm qm | = |rn ||qn − qm | + |qm ||rn − rm | ε ε +M = ε, s QBSB BMHÙO SBDJPOBM QPTJUJWP s Z QBSB UPEP n TVàDJFOUFNFOUF HSBOEF Z QPS MP UBOUP MB TVDFTJÓO {|rn | + rn } FT OVMB  &O DPOTFDVFODJB [{|rn |}] = [{−rn }] = −α = |α|.  &M TJHVJFOUF UFPSFNB SFTVNF MBT QSJODJQBMFT QSPQJFEBEFT EFM WBMPS BCTPMVUP 5FPSFNB  4FBO α, β Z ε OÙNFSPT SFBMFT &OUPODFT B

|α| ≥ 0.

C

|α| = 0 ⇐⇒ α = 0

D

| − α| = |α| 1BSJEBE EFM WBMPS BCTPMVUP 

E

−|α| ≤ α ≤ |α|

F

4FB ε > 0 &OUPODFT |α| = ε ⇐⇒ (α = ε

∨

α = −ε).



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

G

α2 = β 2 ⇐⇒ |α| = |β|

H

|α ± β| ≤ |α| + |β| %FTJHVBMEBE USJBOHVMBS 

I

|α| − |β| ≤ |α ± β|.

J

|α| < ε ⇐⇒ −ε < α < ε

K

L

|α| > ε ⇐⇒ ( α > ε ∨ α < −ε )     |α| − |β| ≤ |α − β|.

M

|αβ| = |α||β|

N

   α  |α| 4J β =  0 FOUPODFT   =  β |β|

%FNPTUSBDJÓO 4FBO α = [{rn }] Z β = [{qn }] B $PNP FTUB QSPQJFEBE WBMF FO FM DBNQP Q FOUPODFT |rn | ≥ 0 QBSB UPEP n Z QPS MPT -FNBT  Z  TF DPODMVZF RVF |α| ≥ 0. C

D Z F TF TJHVFO EF MB EFàOJDJÓO EF WBMPS BCTPMVUP E 1VFTUP RVF −|rn | ≤ rn ≤ |rn | ∀n, OVFWBNFOUF MPT -FNBT  Z  OPT EBO MB QSVFCB EF MB QSPQJFEBE G *HVBM B MB IFDIB FO F EF MB 1SPQPTJDJÓO  H 6TBOEP MB EFTJHVBMEBE |rn ±qn | ≤ |rn |+|qn |, WÃMJEB FO MPT SBDJPOBMFT Z MPT MFNBT DJUBEPT BOUFSJPSNFOUF TF DPODMVZF MB QSVFCB I 4F TJHVF EF H

QVFTUP RVF |α| = |(α ± β) ∓ β| ≤ |α ± β| + |β| J 3FQFUJS MB EFNPTUSBDJÓO EF I FO MB 1SPQPTJDJÓO  K 4F TJHVF EF J

ZB RVF (p ⇐⇒ q) ⇐⇒ (∼ p ⇐⇒ ∼ q). 6TBS UBNCJÊO F  L 6TBS I EPT WFDFT JOUFSDBNCJBOEP MPT SPMFT EF α Z β. M &T DPOTFDVFODJB EF MPT MFNBT Z EF MB JHVBMEBE |rn qn | = |rn ||qn |. N 4F PCUJFOF EF M

ZB RVF |α| = |( αβ )β| = | αβ ||β|.

 

 4VDFTJPOFT EF OÙNFSPT SFBMFT &O FTUB TFDDJÓO FYUFOEFNPT MPT DPODFQUPT QSFTFOUBEPT FO FM DBQÎUVMP  FO DVBOUP B TVDFTJPOFT TF SFàFSF -BT NJTNBT EFàOJDJPOFT EBEBT QB SB OPDJPOFT SFMBDJPOBEBT DPO TVDFTJPOFT FO Q TF UJFOFO QBSB OPDJP OFT DPSSFTQPOEJFOUFT FO FM DBNQP R CBTUB TJNQMFNFOUF QBSBGSBTFBS MB



-PT OÙNFSPT SFBMFT

SFTQFDUJWB EFàOJDJÓO TVTUJUVZFOEP SBDJPOBMFT QPS SFBMFT 1PS FKFNQMP VOB TVDFTJÓO EF OÙNFSPT SFBMFT {αn } FT VOB TVDFTJÓO EF $BVDIZ TJ EB EP ε ∈ R+ FYJTUF N ∈ N UBM RVF TJ m, n > N, FOUPODFT |αn − αm | < ε. 4F EFKB DPNP FKFSDJDJP QBSB FM MFDUPS FTDSJCJS MBT EFNÃT EFàOJDJPOFT -PT UFPSFNBT QSPQPTJDJPOFT Z PCTFSWBDJPOFT QSFTFOUBEPT FO MB TFDDJÓO  UBNCJÊO UJFOFO WBMJEF[ DVBOEP MBT TVDFTJPOFT TPO EF OÙNFSPT SFBMFT MBT QSVFCBT QSFTFSWBO FYBDUBNFOUF MBT NJTNBT JEFBT 0CTFSWBDJÓO :B TBCFNPT RVF TJ q FT VO SBDJPOBM QPEFNPT WFSMP UBN CJÊO DPNP OÙNFSP SFBM B USBWÊT EFM JTPNPSàTNP j : q → [{q}] EFàOJEP FO MB TFDDJÓO BOUFSJPS *HVBM PDVSSF QBSB MBT TVDFTJPOFT EF $BVDIZ EF SBDJPOBMFT FT EFDJS TJ {rn } FT VOB TVDFTJÓO EF $BVDIZ EF SBDJPOBMFT FOUPODFT UBNCJÊO FT VOB TVDFTJÓO EF $BVDIZ EF SFBMFT &O FGFDUP EBEP ε > 0 QPS MB 1SPQPTJDJÓO  FYJTUF s ∈ Q+ DPO s < ε "EFNÃT FYJTUF N0 UBM RVF TJ m, n > N0 FOUPODFT |rn −rm | < s. -VFHP QBSB m, n > N0 | [{rn }] − [{rm }] | = |rn − rm | < s < ε. &M QSJNFS UFPSFNB EF FTUB TFDDJÓO OPT QFSNJUF DPOUJOVBS IBCMBOEP EF VO OÙNFSP SFBM TJO UFOFS RVF VTBS SFQSFTFOUBOUFT EF DMBTF EF FRVJWBMFODJB NÃT QSFDJTBNFOUF DBEB SFBM TFSÃ FM MÎNJUF EF UPEBT MBT TVDFTJPOFT FO Q RVF FTUÃO FO TV DMBTF "EFNÃT DPOUJOVBSFNPT VTBOEP DVBMRVJFS UJQP EF MFUSB QBSB EFOPUBS VO OÙNFSP SFBM 5FPSFNB  4FB α VO SFBM 4J {rn } ∈ α FOUPODFT rn → α FO R. %FNPTUSBDJÓO 4FB ε > 0 1PS MB 1SPQPTJDJÓO  FYJTUF s ∈ Q+ DPO s < ε $PNP {rn } FT EF $BVDIZ FYJTUF n0 UBM RVF TJ m, n > n0 FOUPODFT |rn − rm | < s. &O QBSUJDVMBS TF UJFOF RVF rn − rm < s

Z rm − rn < s,



QBSB UPEP m, n > n0  "IPSB CJFO àKBOEP MB WBSJBCMF n DPO n > n0 FT DMBSP RVF [{rn }] − α = [{rn − rm }∞ m=0 ]. -VFHP VTBOEP MB QSJNFSB EFTJHVBMEBE EF  Z FM -FNB  TF JOàFSF RVF [{rn }] − α ≤ s MP DVBM FT WÃMJEP QBSB DBEB n > n0 . 3FQJUJFOEP MPT NJTNPT BSHVNFOUPT QBSB MB TVDFTJÓO {rm − rn }∞ m=0 TF UJFOF RVF α − [{rn }] ≤ s QBSB DBEB n > n0 . &TUBT EPT ÙMUJNBT EFTJHVBMEBEFT EJDFO RVF | [{rn }] − α| ≤ s ∀n > n0 , MP DVBM JNQMJDB RVF |rn − α| < ε ∀n > n0  Z BTÎ rn → α FO R.  0CTFSWBDJÓO /PUFNPT RVF EF FTUF UFPSFNB TF EFTQSFOEF RVF EBEP α ∈ R FYJTUF VOB TVDFTJÓO EF SBDJPOBMFT {rn } RVF DPOWFSHF B α.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFSDJDJP  %FNVFTUSF VTBOEP MB EFàOJDJÓO RVF MBT TJHVJFOUFT TV DFTJPOFT DPOWFSHFO

 n B

n6 − 5n3 − 2n − 1 

2n7 + 5n C

3n7 − 5n2 + 12 

5 − 2n − 3n2 D

n2 − 4n + 8 

2 n −n+1 . E

2n3 − n2 − 1 &KFSDJDJP  $BMDVMF MÎN

n→∞

1+22 +32 +···+n2 · n3

%FàOJDJÓO  4FBO a Z b SFBMFT àKPT DPO a < b. 4F EFàOFO MPT TJ HVJFOUFT DPOKVOUPT MMBNBEPT JOUFSWBMPT EF OÙNFSPT SFBMFT (a, b) := {x ∈ R : a < x < b} *OUFSWBMP BCJFSUP  [a, b] := {x ∈ R : a ≤ x ≤ b} *OUFSWBMP DFSSBEP  (a, b] := {x ∈ R : a < x ≤ b}; [a, b) := {x ∈ R : a ≤ x < b}; (a, a] = [a, a) = (a, a) := ∅ Z [a, a] := {a}. &TUPT JOUFSWBMPT TF EJDFO BDPUBEPT Z EF MPOHJUVE b − a. 'JKFNPT a ∈ R Z TFB A = {x ∈ R : x > a}. 1PS MB QSPQJFEBE BSRVJNFEJBOB A =  ∅ 'JKFNPT a0 ∈ A. "àSNBNPT RVF A OP FT BDPUBEP TVQFSJPSNFOUF QVFT EF MP DPOUSBSJP FYJTUJSÎB VOB DPOTUBOUF SFBM C UBM RVF x ≤ C QBSB DBEB x ∈ A. $PNP C + 1 > C ≥ a0 > a FOUPODFT C + 1 ∈ A Z FO DPOTFDVFODJB C + 1 ≤ C RVF FT VO BCTVSEP 4JNJMBSFT BàSNBDJPOFT QVFEFO EFDJSTF EFM DPOKVOUP {x ∈ R : x < a} &TUP OPT QFSNJUF JOUSPEVDJS MPT TÎNCPMPT −∞, +∞ Z MPT TJHVJFOUFT DPOKVOUPT MMBNBEPT JOUFSWBMPT OP BDPUBEPT %FàOJDJÓO  4FB a ∈ R àKP (a, +∞) := {x ∈ R : x > a}; [a, +∞) := {x ∈ R : x ≥ a}; (−∞, a) := {x ∈ R : x < a}; (−∞, a] := {x ∈ R : x ≤ a}.



-PT OÙNFSPT SFBMFT

%F FTUB EFàOJDJÓO Z MB USJDPUPNÎB FO R TF EFEVDF RVF (−∞, a) ∪ [a, +∞) = R, MP DVBM OPT JOEVDF B EFàOJS (−∞, +∞) := R. %FàOJDJÓO  %FDJNPT RVF VOB TVDFTJÓO EF OÙNFSPT SFBMFT {an } FT DSFDJFOUF TJ QBSB DBEB n OBUVSBM TF UJFOF RVF an ≤ an+1 . 4J MB EFTJHVBM EBE FT FTUSJDUB EFDJNPT RVF MB TVDFTJÓO FT FTUSJDUBNFOUF DSFDJFOUF -B MMBNBSFNPT EFDSFDJFOUF TJ an ≥ an+1 QBSB DBEB n Z NPOÓUPOB TJ P CJFO FT DSFDJFOUF P CJFO EFDSFDJFOUF -BT OPDJPOFT FTUSJDUBNFOUF EFDSFDJFOUF Z FTUSJDUBNFOUF NPOÓUPOB TF EFàOFO EF NBOFSB TJNJMBS " DPOUJOVBDJÓO QSFTFOUBNPT VO JNQPSUBOUF SFTVMUBEP TPCSF MPT OÙ NFSPT SFBMFT &TUF IFDIP EJTUJOHVF B MPT OÙNFSPT SFBMFT EF MPT OÙNFSPT SBDJPOBMFT QVFT ZB TBCFNPT RVF BVORVF Q FT VO DBNQP PSEFOBEP Z BSRVJNFEJBOP OP FT DPNQMFUP &TUB JODPNQMFUF[ TF USBEVDF FO UÊSNJOPT HFPNÊUSJDPT FO MP TJHVJFOUF TJ DBEB OÙNFSP SBDJPOBM TF JEFOUJàDB DPO VO ÙOJDP QVOUP EF VOB NJTNB SFDUB FTUB RVFEB DPO i IVFDPT u RVF TPO PSJHJOBEPT QPS MB OP FYJTUFODJB FO Q EF MPT TVQSFNPT EF BMHVOPT TVC DPOKVOUPT EF SBDJPOBMFT OP WBDÎPT Z BDPUBEPT TVQFSJPSNFOUF &M QSÓYJNP UFPSFNB EJDF RVF FTUP OP TVDFEF DPO MPT OÙNFSPT SFBMFT B DBEB OÙNFSP SFBM MF DPSSFTQPOEF VO ÙOJDP QVOUP FO MB SFDUB Z WJDFWFSTB OP RVFEBO iIVFDPTu 5FPSFNB  "YJPNB EF DPNQMFUF[ EF R  &M DBNQP R EF MPT OÙNFSPT SFBMFT FT DPNQMFUP %FNPTUSBDJÓO 4FB A ⊂ R OP WBDÎP Z BDPUBEP TVQFSJPSNFOUF 1SPCFNPT RVF A UJFOF TVQSFNP FO R. 7BNPT B DPOTFHVJS VO JOUFSWBMP FO FM DVBM iDBQUVSBSFNPTu FM TVQSFNP -B JEFB EF MB QSVFCB FT DPOTUSVJS VOB TVDF TJÓO EF SBDJPOBMFT MB DVBM TFSÃ EF $BVDIZ Z QPS UBOUP EFUFSNJOBSÃ VO SFBM RVF SFTVMUBSÃ TFS FM TVQSFNP CVTDBEP 4FB s ∈ R VOB DPUB TVQFSJPS EF A $PNP R FT BSRVJNFEJBOP FYJTUF VO OBUVSBM M ≥ 1 UBM RVF s < M Z BTÎ M FT DPUB TVQFSJPS EF A 1PS PUSP MBEP FYJTUFO a ∈ A Z m ∈ N DPO m ≥ 1 UBMFT RVF −a < m. 1PS UBOUP −m < M Z DPO FTUP DPOTFHVJNPT FM JOUFSWBMP DFSSBEP [−m, M ]. "IPSB QBSB DBEB p ∈ N DPOTJEFSFNPT FO USF −m Z M MPT (M + m)2p + 1 OÙNFSPT SFBMFT EF MB GPSNB −m + k · 2−p DPO k = 0, 1, . . . , (M + m)2p  4FB rp FM NFOPS OBUVSBM UBM RVF rp ap := −m + p 2 FT DPUB TVQFSJPS EF A 5BM OBUVSBM FYJTUF QVFT DPO rp = (M + m)2p PCUFOFNPT ap = M. /PUFNPT RVF QBSB UPEP p ∈ N ap −

rp − 1 1 = −m + p 2 2p 

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

OP FT DPUB TVQFSJPS EF A QVFT rp FT FM NFOPS OBUVSBM DPO FTUB QSPQJFEBE -B TVDFTJÓO {ap } SFTVMUB TFS EFDSFDJFOUF ap+1 − ap =

rp+1 − 2rp · 2p+1

4J FM OVNFSBEPS FO MB BOUFSJPS JHVBMEBE GVFSB QPTJUJWP QBSB BMHÙO OBUVSBM p FTUP FT rp+1 > 2rp FOUPODFT rp+1 − 1 ≥ 2rp  FTUP JNQMJDBSÎB RVF ap+1 −

1 2p+1

≥ ap

Z TFSÎB DPUB TVQFSJPS EF A QVFT ap MP FT MP DVBM FT JNQPTJCMF 1PS UBOUP EJDIP OVNFSBEPS FT OP QPTJUJWP QBSB UPEP p ∈ N Z EF FTUB NBOFSB TF UJFOF RVF ap+1 ≤ ap  .ÃT BÙO UFOFNPT RVF TJ q > p, FOUPODFT aq ≤ ap  "EFNÃT DPNP ap − 21p OP FT DPUB TVQFSJPS EF A FYJTUF xp ∈ A UBM RVF ap − 21p < xp . -VFHP 1 

ap − p < aq ≤ ap , 2 EF EPOEF FODPOUSBNPT RVF |aq − ap | < 2−p QBSB UPEP QBS EF OBUVSBMFT p Z q DPO p < q. &TUB EFTJHVBMEBE OPT QFSNJUF DPODMVJS RVF MB TVDFTJÓO {ap } FT EF $BVDIZ FO Q -VFHP FTUB TVDFTJÓO EFUFSNJOB VO OÙNFSP SFBM EJHBNPT α Z QPS FM UFPSFNB BOUFSJPS UFOFNPT RVF ap → α DVBOEP p → ∞. )BDJFOEP RVF q → ∞ PCUFOFNPT EF  Z FM -FNB  ap −

1 ≤ α ≤ ap , 2p



MP DVBM FT WÃMJEP QBSB UPEP p. 'JOBMNFOUF QSPCFNPT RVF α = TVQ A DPO MP RVF UFSNJOBNPT MB QSVFCB EFM UFPSFNB 7FBNPT QSJNFSP RVF α FT DPUB TVQFSJPS EF A TJ OP GVFSB BTÎ FYJTUJSÎB a ∈ A UBM RVF a > α $PNP R FT BSRVJNFEJBOP UFOESÎBNPT VO OBUVSBM po UBM RVF 2p1o < a − α. $PO FTUP TF TFHVJSÎB FO QBSUJDVMBS EF 

a p0 − α ≤

1 < a − α, 2 p0

Z TF DPODMVJSÎB RVF ap0 < a MP RVF OPT EJSÎB RVF ap0 OP FT DPUB TVQFSJPS EF A &TUB DPOUSBEJDDJÓO NVFTUSB RVF α FT DPUB TVQFSJPS EF A 4VQPOHB NPT BIPSB RVF FYJTUF VOB DPUB TVQFSJPS s EF A DPO s < α. $PNP BOUFT FODPOUSBNPT VO OBUVSBM p1 UBM RVF 1 < α − s . 2p1 

-PT OÙNFSPT SFBMFT

"IPSB CJFO DPNP ap1 − RVF

1 2 p1

OP FT DPUB TVQFSJPS EF A FYJTUF a ˆ ∈ A UBM a ˆ > ap1 −

1 2 p1

Z FO DPOTFDVFODJB ˆ+ a p1 < a

1 1  ≤ s + < α, p 21 2p1

FTUP FT ap1 < α DPOUSBEJDJFOEP   1PS UBOUP α ≤ s QBSB UPEB DPUB   TVQFSJPS s EF A Z EF FTUB NBOFSB α = TVQ A. 0CTFSWBDJÓO 4F QVFEF EFNPTUSBS RVF DVBMRVJFS DBNQP PSEFOBEP Z DPNQMFUP FT JTPNPSGP B MPT OÙNFSPT SFBMFT QPS FKFNQMP  1PS UBOUP QPEFNPT BàSNBS RVF FYJTUF VO ÙOJDP DBNQP PSEFOBEP Z DPNQMFUP B TBCFS FM DBNQP EF MPT OÙNFSPT SFBMFT -B TJHVJFOUF EFTJHVBMEBE EFTFNQFÒBSÃ VO QBQFM JNQPSUBOUF FO QSÓ YJNBT EFNPTUSBDJPOFT TF MMBNB EFTJHVBMEBE EF #FSOPVMMJ 1SPQPTJDJÓO  4FB a ∈ R DPO a ≥ −1 àKP &OUPODFT QBSB DBEB OBUVSBM n ≥ 1 TF UJFOF RVF (1 + a)n ≥ 1 + na. %FNPTUSBDJÓO 4J n = 1 MB BàSNBDJÓO TF DVNQMF 4VQPOHBNPT RVF QBSB n ≥ 1 TF WFSJàDB MB EFTJHVBMEBE Z QSPCÊNPTMB QBSB n + 1. (1 + a)n+1 = (1 + a)n (1 + a) ≥ (1 + na)(1 + a) = 1 + (1 + n)a + na2 ≥ 1 + (n + 1)a, MP RVF EFNVFTUSB MB QSPQPTJDJÓO QBSB UPEP OBUVSBM n ≥ 1.

 

5FPSFNB  5FPSFNB EFM TBOEXJDI  4FBO {xn }, {yn } Z {zn } TVDF TJPOFT EF OÙNFSPT SFBMFT UBMFT RVF FYJTUF N0 ∈ N DPO xn ≤ zn ≤ yn ∀n ≥ N0 . 4J xn → L ∈ R Z yn → L, FOUPODFT zn → L. %FNPTUSBDJÓO 4FB ε > 0 1PS MB DPOWFSHFODJB EF MBT TVDFTJPOFT {xn } Z {yn } FYJTUF VO OBUVSBM N1 ≥ N0 UBM RVF |xn − L| < ε

Z

|yn − L| < ε

TJ

n ≥ N1 .

"IPSB CJFO TJ n ≥ N1 FOUPODFT zn − L = (zn − yn ) + (yn − L) ≤ |yn − L| < ε.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

1PS PUSB QBSUF TJ n ≥ N1 FOUPODFT zn − L = (zn − xn ) + (xn − L) ≥ −|xn − L| > −ε. %F FTUP TF TJHVF RVF TJ n ≥ N1 FOUPODFT |zn − L| < ε MP DVBM EFNVFTUSB  FM UFPSFNB  n 

1 · %FNVFTUSF RVF MB TVDFTJÓO {qn } k + n2 k=1 DPOWFSHF y2VÊ TF QVFEF BàSNBS EF {nqn } &KFSDJDJP  4FB qn =

&KFNQMP  4FB b ∈ R DPO |b| < 1 &OUPODFT MB TVDFTJÓO {bn }∞ n=1 FT UBM RVF bn → 0. 1BSB b = 0 FT JONFEJBUB MB BàSNBDJÓO BTÎ RVF QPEFNPT TVQPOFS RVF 0 < b < 1 QVFT FO FM PUSP DBTP TF BQMJDB FM SFTVMUBEP B 0 < −b < 1. -B EFTJHVBMEBE EF #FSOPVMMJ OPT QSPQPSDJPOB MPT TJHVJFOUFT DÃMDVMPT   1 1−b n n = 1+ ≥ 1 + (1 − b) > n(1 − b), n b b b QBSB UPEP n ∈ N, n ≥ 1 Z QPS UBOUP 0 < bn <

1 n(1 − b)

QBSB DBEB n ≥ 1 MP DVBM JNQMJDB RVF bn → 0. 5FPSFNB  4J {an } FT VOB TVDFTJÓO EF SFBMFT BDPUBEB TVQFSJPSNFOUF Z DSFDJFOUF FOUPODFT {an } DPOWFSHF %FNPTUSBDJÓO "QMJDBNPT FM BYJPNB EF DPNQMFUF[ EF R FO FTUF DBTP BM DPOKVOUP A := {an : n ∈ N}, RVF DMBSBNFOUF FT OP WBDÎP Z BDPUBEP TVQFSJPSNFOUF 4FBO a := TVQ A Z ε > 0 -VFHP FYJTUF VO OBUVSBM N UBM RVF a−ε < aN Z QPS MB NPOPUPOÎB EF {an } a − ε < aN ≤ an ≤ a < a + ε, QBSB UPEP n ≥ N. 1PS UBOUP an → a.

 

/PUFNPT RVF FM UFPSFNB EJDF RVF VOB TVDFTJÓO {an } DSFDJFOUF Z BDP UBEB TVQFSJPSNFOUF DPOWFSHF BM OÙNFSP SFBM TVQ{an : n ∈ N}. $PSPMBSJP  4J {an } FT VOB TVDFTJÓO EF SFBMFT BDPUBEB JOGFSJPSNFOUF Z EFDSFDJFOUF FOUPODFT {an } DPOWFSHF &O FTUF DBTP an → ÎOG {an : n ∈ N}.



-PT OÙNFSPT SFBMFT

$PSPMBSJP  5PEB TVDFTJÓO {an } EF SFBMFT NPOÓUPOB Z BDPUBEB DPO WFSHF %FàOJDJÓO  6OB TVDFTJÓO EF JOUFSWBMPT {In } TF MMBNB TVDFTJÓO EF JOUFSWBMPT FODBKBEPT TJ In+1 ⊆ In QBSB DBEB n ∈ N. 5FPSFNB  %F MPT JOUFSWBMPT FODBKBEPT  4FB In = [an , bn ] DPO an < bn , VOB TVDFTJÓO EF JOUFSWBMPT FODBKBEPT &OUPODFT I :=

∞ 

In = ∅.

n=0

"EFNÃT TJ MB MPOHJUVE EF In = bn − an → 0, EJDIB JOUFSTFDDJÓO FT VO DPOKVOUP TJOHVMBS %FNPTUSBDJÓO $PNP MPT JOUFSWBMPT TPO FODBKBEPT FOUPODFT an ≤ an+1 Z bn+1 ≤ bn QBSB UPEP n Z EF FTUP TF UJFOF RVF a0 ≤ an < bn ≤ b0 QBSB UPEP n 1PS MPT SFTVMUBEPT QSFWJPT MBT TVDFTJPOFT {an } Z {bn } TPO DPOWFSHFOUFT DPO MÎNJUFT EJHBNPT a Z b UBMFT RVF an ≤ a Z b ≤ bn  NÃT BÙO a ≤ b &O DPOTFDVFODJB a, b ∈ I Z BEFNÃT an −bn ≤ a−b ≤ bn −an QBSB UPEP n EF EPOEF TF EFEVDF RVF a = b TJ bn − an → 0 &TUP JNQMJDB RVF a ∈ I 'JOBMNFOUF FT TFODJMMP WFS RVF I ⊆ {a} Z DPO FTUP TF  EFNVFTUSB RVF I = {a}.  0CTFSWBDJÓO &O FTUF UFPSFNB FT DSVDJBM MB IJQÓUFTJT EF RVF MPT JOUFS WBMPT FODBKBEPT In TFBO DFSSBEPT Z BDPUBEPT DPNP MP EFNVFTUSBO MBT TJHVJFOUFT TVDFTJPOFT EF JOUFSWBMPT FODBKBEPT

  1  0, Z { [n, ∞) } , n+1 QBSB MBT DVBMFT I = ∅. &KFSDJDJP  4FB {αn } VOB TVDFTJÓO EF OÙNFSPT SFBMFT UBM RVF |αn − αn+1 | <

1 ∀ n ≥ 1. n(n + 1)

1SVFCF RVF {αn } FT EF $BVDIZ n  1 . 1SPCFNPT &KFNQMP  4F EFàOF MB TVDFTJÓO EF SFBMFT QPS an = k2 k=1 RVF {an } FT VOB TVDFTJÓO EF $BVDIZ -B QSPQJFEBE EF MB TVNB UFMFTDÓQJDB



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

FT EF HSBO BZVEB FO FTUF FKFNQMP &M QBTP DMBWF FTUÃ BM JOJDJP QBSB n > m > 1 TF UJFOF RVF |an − am | = =

n 

n  1 1 < 2 k k(k − 1)

k=m+1 n   k=m+1

k=m+1

1 1 1 1 1 − = − < · k−1 k m n m

"IPSB TÎ EBEP ε > 0 FYJTUF N ∈ N UBM RVF 1/N < ε -VFHP TJ m > n ≥ N FOUPODFT |an − am | < ε MP DVBM EFNVFTUSB RVF MB TVDFTJÓO EBEB FT EF $BVDIZ n  1 &KFSDJDJP  4F EFàOF MB TVDFTJÓO EF SFBMFT QPS an = . 1SVFCF k3 k=1 RVF {an } FT VOB TVDFTJÓO EF $BVDIZ

&KFSDJDJP  %FNVFTUSF RVF MB TVDFTJÓO n  1 DPOWFSHF Z IBMMF TV MÎNJUF an = 2 k + 2k

{an }

EBEB

QPS

k=1

&KFSDJDJP  $BMDVMF FM MÎNJUF EF MB TVDFTJÓO {(2n + 3n )1/n }. 5FPSFNB  5PEB TVDFTJÓO EF $BVDIZ EF OÙNFSPT SFBMFT DPOWFSHF FO R %FNPTUSBDJÓO 4FB {xn } VOB TVDFTJÓO EF $BVDIZ FO R -VFHP FYJTUF VOB DPOTUBOUF M > 0 UBM RVF |xn | ≤ M Z QPS UBOUP QBSB DBEB OBUVSBM n UJFOF TFOUJEP EFàOJS yn := TVQ{xn , xn+1 , xn+2 , . . .} = TVQ{xk : k ≥ n}. $PNP {xn+1 , xn+2 , . . .} ⊆ {xn , xn+1 , xn+2 , . . .} FOUPODFT yn+1 ≤ yn SFTVMUBOEP BTÎ RVF MB TVDFTJÓO {yn } FT EFDSFDJFOUF Z BEFNÃT FT BDPUBEB JOGFSJPSNFOUF QVFTUP RVF yn ≥ xn ≥ −M QBSB UPEP OBUVSBM n. &O DPOTFDVFODJB FYJTUF y ∈ R UBM RVF yn −→ y "àSNBNPT RVF xn −→ y &O FGFDUP TFB ε > 0 &YJTUF VO OBUVSBM No UBM RVF |yn − y| < ε/3, TJ n ≥ No .



1PS TFS {xn } EF $BVDIZ FYJTUF N1 ∈ N EF NBOFSB RVF |xn − xm | < ε/3 TJ n, m ≥ N1 .





-PT OÙNFSPT SFBMFT

4FB N2 = NÃY{No , N1 }. 1PS DBSBDUFSJ[BDJÓO EF TVQSFNP FYJTUF VO OB UVSBM N3 ≥ N2 UBM RVF yN2 − ε/3 < xN3 ≤ yN2 , MP RVF JNQMJDB |yN2 − xN3 | < ε/3.



'JOBMNFOUF QBSB n ≥ N3 Z UFOJFOEP FO DVFOUB 

 Z  PCUF OFNPT RVF |y − xn | ≤ |y − yN2 | + |yN2 − xN3 | + |xN3 − xn | < ε/3 + ε/3 + ε/3 = ε. &TUP EFNVFTUSB MB BàSNBDJÓO Z FM UFPSFNB

 

0CTFSWBDJÓO -B TVDFTJÓO {yn } Z FM SFBM y QSFTFOUBEPT FO MB EFNPTUSB DJÓO EFM UFPSFNB BOUFSJPS EFàOFO FM MÎNJUF TVQFSJPS EF MB TVDFTJÓO {xn } .ÃT QSFDJTBNFOUF FM MÎNJUF TVQFSJPS EF VOB TVDFTJÓO {xn } EFOPUBEP QPS MÎN TVQ xn , n→∞

TF EFàOF DPNP MÎN yn EPOEF n→∞

yn := TVQ{xn , xn+1 , xn+2 , . . .} = TVQ{xk : k ≥ n}. 4JNJMBSNFOUF TF EFàOF FM MÎNJUF JOGFSJPS EF VOB TVDFTJÓO {xn } EFOPUBEP QPS MÎN JOG xn , n→∞

DPNP MÎN zn EPOEF zn := ÎOG {xn , xn+1 , xn+2 , . . .} = ÎOG {xk : k ≥ n} n→∞

-PT 5FPSFNBT  Z  OPT QSPQPSDJPOBO FM TJHVJFOUF SFTVMUBEP 5FPSFNB  R FT DPNQMFUP TJ Z TPMP TJ UPEB TVDFTJÓO EF $BVDIZ EF OÙNFSPT SFBMFT DPOWFSHF FO R. 4BCFNPT RVF FO R UPEB TVDFTJÓO FT DPOWFSHFOUF TJ Z TPMP TJ FT EF $BVDIZ -B TVàDJFODJB FT DJFSUB TPMBNFOUF FO MPT iFTQBDJPTu RVF TPO DPNQMFUPT &M DPOKVOUP EF MPT OÙNFSPT SBDJPOBMFT FT VO FKFNQMP FO FM DVBM FYJTUFO TVDFTJPOFT EF $BVDIZ RVF OP TPO DPOWFSHFOUFT FO EJDIP DPOKVOUP JODPNQMFUF[  &KFNQMP  $POTJEFSFNPT MB TVDFTJÓO EF SBDJPOBMFT EFàOJEB EF NB OFSB SFDVSTJWB BTÎ q1 = 1 qn+1 = 12 (qn + q2n ), TJ n ≥ 1. 

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

7FBNPT RVF FTUB TVDFTJÓO FT EF $BVDIZ FO Q QFSP OP DPOWFSHF FO Q 2 ) EF MP DVBM TF EFTQSFOEF 1BSB n ≥ 2 UFOFNPT RVF qn = 12 (qn−1 + qn−1 RVF 0 ≤ (qn−1 − qn )2 = qn2 − 2. 

&TUP JNQMJDB RVF qn2 ≥ 2 QBSB UPEP OBUVSBM n ≥ 2. 1PS PUSB QBSUF FT JONFEJBUP RVF q2 − 2 |qn+1 − qn | = n . 

2qn $PNCJOBOEP  Z  PCUFOFNPT MB JHVBMEBE |qn+1 − qn | =

1 |qn−1 − qn |2 , 2qn

∀ n ≥ 2.



"EFNÃT UFOJFOEP FO DVFOUB RVF 1 2 q2 + 2 q2 + 1 1 1 qn+1 = (qn + ) = n = n + ≥1+ 2 qn 2qn 2qn 2qn 2qn Z VO BSHVNFOUP EF JOEVDDJÓO TF NVFTUSB RVF qn ≥ 1 QBSB UPEP OBUVSBM n -MFWBOEP FTUB DPODMVTJÓO B  TF MMFHB B MB EFTJHVBMEBE 1 |qn+1 − qn | ≤ |qn − qn−1 |2 , 2

∀ n ≥ 2,

MB DVBM OPT QFSNJUF EFNPTUSBS WÎB JOEVDDJÓO NBUFNÃUJDB FM TJHVJFOUF FTUJNBUJWP  2n −1  2n  2n  n 1 1 1 1 =2 n, |qm − qn | = |

m−1 

m−1 

i=n m−1 

i=n

(qi+1 − qi )| ≤

|qi+1 − qi |

 1 i ≤2 (QPS  ) 4 i=n  m−n     8 1 n 1 = 1− 3 4 4 <



8 . 3 · 4n

-PT OÙNFSPT SFBMFT

&TUBNPT ZB DBTJ MJTUPT QBSB MB DPODMVTJÓO 4FB ε ∈ Q DPO ε > 0. 1PS MB QSPQJFEBE BSRVJNFEJBOB FYJTUF VO OBUVSBM N UBM RVF 41N < 3ε 8 . %F FTUB NBOFSB TJ m, n TPO OBUVSBMFT DPO m > n ≥ N FOUPODFT |qm − qn | <

8 8 ≤ < ε, n 3·4 3 · 4N

MP RVF EFNVFTUSB RVF MB TVDFTJÓO {qn } FT EF $BVDIZ FO Q. 'JOBMNFOUF WFBNPT RVF EJDIB TVDFTJÓO OP DPOWFSHF FO Q 4VQPOHBNPT MP DPOUSBSJP RVF FYJTUF q ∈ Q UBM RVF qn → q. &T TFODJMMP WFS RVF qn+1 → q. -VFHP BM UPNBS MÎNJUF DVBOEP n → ∞ FO MB JHVBMEBE RVF EFàOF qn , PCUFOFNPT RVF 1 2 q = (q + ) ⇐⇒ q 2 = 2, 2 q MP RVF FT VOB DPOUSBEJDDJÓO QVFT OP FYJTUF VO SBDJPOBM DVZP DVBESBEP TFB  &KFSDJDJP  6OB TVDFTJÓO EF OÙNFSPT SFBMFT {an } TF EJDF RVF FT DPOUSBDUJWB TJ FYJTUF VOB DPOTUBOUF C < 1 UBM RVF QBSB UPEP OBUVSBM n TF DVNQMF RVF |an+2 − an+1 | ≤ C |an+1 − an |. 1SVFCF RVF UPEB TVDFTJÓO DPOUSBDUJWB FT EF $BVDIZ &KFSDJDJP  4F EFàOF MB TVDFTJÓO EF SFBMFT EF NBOFSB SFDVSTJWB BTÎ 1 x1 = 1, x2 = 2, xn = (xn−1 + xn−2 ) QBSB n ≥ 3. 2 1SVFCF RVF {xn } FT VOB TVDFTJÓO EF $BVDIZ 4VHFSFODJB IBMMF VOB GÓSNVMB QBSB |xn − xn+1 |. &KFSDJDJP  4FBO {an }, {bn } TVDFTJPOFT EF OÙNFSPT SFBMFT RVF DPO WFSHFO B A Z B SFTQFDUJWBNFOUF 4VQPOHB RVF FYJTUF N0 ∈ N DPO an ≤ bn QBSB UPEP n ≥ N0 . %FNVFTUSF RVF A ≤ B. &O MB TJHVJFOUF QSPQPTJDJÓO QSFTFOUBNPT MB QBSUF FOUFSB EF VO OÙ NFSP SFBM DPODFQUP RVF FT EF NVDIB VUJMJEBE FO UFPSÎB EF OÙNFSPT 7FS QPS FKFNQMP FM UFYUP  1SPQPTJDJÓO  %BEP x ∈ R FYJTUF VO ÙOJDP FOUFSP Nx UBM RVF Nx ≤ x < Nx + 1. 5BM FOUFSP TF MMBNB QBSUF FOUFSB EF x Z TF EFOPUB QPS [[ x ]], FT EFDJS [[ x ]] ≤ x < [[ x ]] + 1.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

%FNPTUSBDJÓO 4J x ∈ Z FT DMBSP RVF Nx = x TBUJTGBDF MB DPODMVTJÓO EFM UFPSFNB 4VQPOHBNPT RVF x ∈ / Z Z BEFNÃT RVF x > 0 4FB A ≡ Ax = {n ∈ N : x < n}. 1PS MB QSPQJFEBE BSRVJNFEJBOB A = ∅ Z QPS FM 1#0 FYJTUF m ≡ mx = NÎO A ≥ 1. $PNP m−1 ≥ 0 FT VO OBUVSBM Z x OP FT VO FOUFSP FOUPODFT QPS MB NJOJ NBMJEBE UFOFNPT RVF m − 1 < x < m DPODMVZÊOEPTF MB FYJTUFODJB DPO Nx = m − 1. 4VQPOHBNPT BIPSB RVF x < 0 1PS FM SB[POBNJFOUP QSFWJP BQMJDBEP B −x FYJTUF VO OBUVSBM M ≥ 1 UBM RVF M − 1 < −x < M EF MP DVBM TF TJHVF RVF −M < x < −M + 1 FT EFDJS UFOFNPT MB FYJT UFODJB DPO Nx = −M. 'JOBMNFOUF QSPCFNPT MB VOJDJEBE TVQPOHBNPT RVF FYJTUF PUSP FOUFSP Kx UBM RVF Kx ≤ x < Kx + 1. $PNCJOBOEP EF TJHVBMEBEFT DPOTFHVJNPT Nx ≤ x < Kx + 1, EF EPOEF TF JOàFSF RVF Nx ≤ Kx ; Z DPNCJOBOEP MBT PUSBT EFTJHVBMEBEFT BOÃMPHBT JOGFSJNPT RVF Kx ≤ Nx , FTUP FT Nx = Kx . &TUP QSVFCB MB VOJDJEBE Z FM UFPSFNB  RVFEB QSPCBEP  0CTFSWFNPT RVF EF MP BOUFSJPS TF EFTQSFOEF RVF x − 1 < [[ x ]] ≤ x. &KFNQMP  &T DMBSP RVF TJ x ∈ Z FOUPODFT [[ x ]] + [[ −x ]] = x − x = 0. "IPSB CJFO TJ x ∈ / Z FOUPODFT BM TVNBS MBT EFTJHVBMEBEFT x − 1 < [[ x ]] < x, Z − x − 1 < [[ −x ]] < −x. PCUFOFNPT −2 < [[ x ]] + [[ −x ]] < 0 %F FTUP TF EFEVDF RVF [[ x ]] + [[ −x ]] = −1. &O SFTVNFO IFNPT QSPCBEP RVF [[ x ]] + [[ −x ]] =

0 TJ x ∈ Z −1 TJ x ∈ / Z.

&KFSDJDJP  4FBO x ∈ R àKP Z A = {m ∈ Z : m ≤ x}. 1SVFCF RVF NÃY A = [[ x ]]. $PODMVZB RVF TJ m ∈ Z FT UBM RVF m ≤ x FOUPODFT m ≤ [[ x ]]. &KFSDJDJP  %FNVFTUSF B m ∈ Z ⇐⇒ [[ x + m ]] = [[ x ]] + m.



-PT OÙNFSPT SFBMFT

C [[ x ]] + [[ y ]] ≤ [[ x + y ]] ≤ [[ x ]] + [[ y ]] + 1. D 1BSB UPEP OBUVSBM n ≥ 1 TF DVNQMF RVF ∀ x ∈ R, [[ nx ]] ≤ [[ nx ]] . $PNP DPOTFDVFODJB TF JOàFSF RVF [[ nx ]] ≥ n[[ x ]] QBSB UPEP OBUVSBM n ≥ 1 Z QBSB UPEP x ∈ R. E 4J x ≤ y FOUPODFT [[ x ]] ≤ [[ y ]]. F 4FBO x, y SFBMFT QPTJUJWPT .VFTUSF RVF [[ x ]] [[ y ]] ≤ [[ xy ]]. G 4FB n ∈ Z &OUPODFT [[ x ]] = n TJ Z TPMP TJ FYJTUF z ∈ [0, 1) UBM RVF x = n + z. &M OÙNFSP SFBM z FT MMBNBEP MB QBSUF GSBDDJPOBSJB EF x RVF EFOPUBNPT QPS ((x)) &KFNQMP  %FNPTUSFNPT RVF TJ n ≥ 1 FT OBUVSBM FOUPODFT [[ nx ]] = [[ [[ x ]]/n ]]. 6OB FTUSBUFHJB RVF VTBSFNPT FT QSPCBS NVUVB EFTJHVBMEBE DPNP [[ x ]] ≤ x FOUPODFT [[ x ]]/n ≤ x/n Z BTÎ QPS FM MJUFSBM E EFM &KFSDJDJP  DPODMVJNPT RVF [[ nx ]] ≥ [[ [[ x ]]/n ]]. 1BSB QSPCBS MB EFTJHVBMEBE JO WFSTB TFB ε > 0 Z àKFNPT UBOUP x DPNP n $PNP [[ nx ]] = NÃY{m ∈ Z : m ≤ x/n} FOUPODFT FYJTUF VO FOUFSP k ≤ x/n UBM RVF [[ nx ]] − ε < k. "IPSB CJFO k ≤ [[ nx ]] Z MB QBSUF D EFM &KFSDJDJP  JNQMJDBO RVF k ≤ [[ nx ]] Z QPS DPOTJHVJFOUF k ≤ [[ [[ x ]]/n ]]. &O DPOTFDVFODJB [[ nx ]] − ε < [[ [[ x ]]/n ]]. 1VFTUP RVF ε > 0 FT BSCJUSBSJP OFDFTBSJBNFOUF [[ nx ]] ≤ [[ [[ x ]]/n ]]. &KFNQMP  %FNPTUSFNPT RVF [[ x ]] + [[ x + 1/2 ]] = [[ 2x ]]. 4FB x = [[ x ]] + z DPO 0 ≤ z < 1 WFS MJUFSBM G EFM FKFSDJDJP BOUFSJPS  $POTJEFSFNPT EPT DBTPT $BTP  0 ≤ z < 1/2 5FOFNPT FOUPODFT RVF x + 1/2 = [[ x ]] + (z + 1/2) DPO 1/2 ≤ z + 1/2 < 1.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-VFHP QPS FM MJUFSBM DJUBEP [[ x + 1/2 ]] = [[ x ]] 5BNCJÊO TF UJFOF RVF 2x = 2[[ x ]] + 2z DPO 0 ≤ 2z < 1 Z QPS UBOUP [[ 2x ]] = 2[[ x ]]. &O DPOTF DVFODJB [[ x ]] + [[ x + 1/2 ]] = [[ 2x ]]. $BTP  1/2 ≤ z < 1 #BKP FTUF TVQVFTUP TF UJFOF RVF x + 1/2 = [[ x ]] + z + 1/2 = [[ x ]] + 1 + (z − 1/2) DPO 0 ≤ z − 1/2 < 1/2. -VFHP [[ x + 1/2 ]] = [[ x ]] + 1. 1PS PUSB QBSUF 2x = 2[[ x ]] + 2z = 2[[ x ]] + 1 + (2z − 1) DPO 0 ≤ 2z − 1 < 1 Z EF FTUB NBOFSB [[ 2x ]] = 2[[ x ]] + 1. 1PS UBOUP [[ x ]] + [[ x + 1/2 ]] = 2[[ x ]] + 1 = [[ 2x ]]. &KFSDJDJP  B 1SVFCF RVF [[ x ]] + [[ x + 1/3 ]] + [[ x + 2/3 ]] = [[ 3x ]]. C 1SVFCF RVF [[ x ]] − 2[[ x/2 ]] ∈ {0, 1}. D .VFTUSF RVF FM OÙNFSP EF FOUFSPT m QBSB MPT DVBMFT x < m ≤ y FT [[ y ]] − [[ x ]]. E 1SVFCF RVF FM SFTUP BM EJWJEJS FM FOUFSP a QPS FM FOUFSP m > 0 FT m((a/m)) EPOEF ((x)) = x − [[ x ]] FT MB QBSUF GSBDDJPOBSJB EFM SFBM x. F 4FBO m ≥ 1 VO OBUVSBM Z x > 0 %FNVFTUSF RVF FM OÙNFSP EF NÙMUJQMPT QPTJUJWPT EF m RVF OP FYDFEFO B x FTUÃ EBEP QPS [[ x/m ]]. 6OB BQMJDBDJÓO JONFEJBUB EF FTUB BàSNBDJÓO FT MB TJHVJFOUF TFBO n Z a FOUFSPT QPTJUJWPT &OUPODFT FM OÙNFSP EF FOUFSPT EF MB TVDF TJÓO 1, 2, . . . , n RVF TPO EJWJTJCMFT QPS a FT [[ n/a ]]. &O FGFDUP FTUP FRVJWBMF B FODPOUSBS FM OÙNFSP EF NÙMUJQMPT EF a RVF OP FYDFEFO B n QPS MP BàSNBEP JOJDJBMNFOUF FTUF OÙNFSP FT [[ n/a ]]. " DPOUJOVBDJÓO QSFTFOUBNPT VO UFPSFNB RVF HBSBOUJ[B MB FYJTUFODJB EF MB SBÎ[ nÊTJNB EF VO OÙNFSP SFBM QPTJUJWP Z TV EFNPTUSBDJÓO FTUÃ TPQPSUBEB FO FM BYJPNB EF DPNQMFUF[ %F FTUF SFTVMUBEP TF EFTQSFOEF MB FYJTUFODJB EF SFBMFT RVF OP TPO SBDJPOBMFT &TUF IFDIP EB BM BYJPNB EF DPNQMFUF[ VOB JNQPSUBODJB OPUBCMF FO NBUFNÃUJDBT QVFT QFSNJUF EFNPTUSBS MB FYJTUFODJB EF MPT OÙNFSPT MMBNBEPT JSSBDJPOBMFT 5FPSFNB  4FBO a ∈ R DPO a > 0 Z n ≥ 1 OBUVSBM àKPT &OUPODFT FYJTUF VO ÙOJDP SFBM QPTJUJWP b UBM RVF bn = a.



-PT OÙNFSPT SFBMFT

%FNPTUSBDJÓO %FàOBNPT FM DPOKVOUP S := {x ∈ R+ : xn < a}. S FT OP WBDÎP QVFTUP RVF FYJTUF VO OBUVSBM N DPO 1/N < a QFSP BEFNÃT FT GÃDJM WFS QPS JOEVDDJÓO RVF N m ≥ N QBSB UPEP m ∈ N DPO m ≥ 1 Z QPS UBOUP 1/N n ≤ 1/N < a MP RVF RVJFSF EFDJS RVF 1/N ∈ S &TUF DPOKVOUP SFTVMUB BDPUBEP TVQFSJPSNFOUF TFB x ∈ S QPS USJDPUPNÎB a > 1 P CJFO a ≤ 1 &O FM QSJNFS DBTP an ≥ a Z BTÎ xn < a ≤ an . %F FTUP TF JOàFSF RVF (x − a)(xn−1 + xn−2 a + xn−3 a2 + · · · + an−1 ) < 0, FT EFDJS x < a. &O FM TFHVOEP DBTP TF EFEVDF JONFEJBUBNFOUF RVF x < 1 &O DPODMVTJÓO UFOFNPT RVF x ≤ NÃY{a, 1} 1PS MB DPNQMFUF[ EF R FYJTUF b ∈ R UBM RVF b = TVQ S, DPO b ≥ 1/N > 0. 7BNPT B QSPCBS RVF bn = a. 1BSB m ≥ N UFOFNPT RVF 1/m ∈ S Z QPS UBOUP 0 ≤ b − 1/m < xm QBSB BMHÙO xm ∈ S -VFHP   1 n b− < xnm < a m TJFNQSF RVF m ≥ N ; FTUP DPNCJOBEP DPO FM &KFSDJDJP  QSPEVDF RVF 1 1 n bn ≤ a. 1PS PUSB QBSUF DPNP b + m ∈ / S TF UJFOF RVF (b + m ) ≥ a n QBSB UPEP m ≥ 1 OBUVSBM Z BTÎ b ≥ a DPNP DPOTFDVFODJB DPODMVJNPT MB JHVBMEBE bn = a. -B VOJDJEBE TF TJHVF EF 0 = bn1 − bn2 = (b1 − b2 )

n 

i−1 bn−i 1 b2 ,



i=1

Z EF RVF

n 

 

i−1 bn−i FT QPTJUJWP 1 b2

i=1

&O FM DBTP a = 0 FT DMBSP RVF b = 0. %FàOJDJÓO  &M SFBM b FO FM UFPSFNB BOUFSJPS DVBOEP n ≥ 2 MP √ MMBNBNPT SBÎ[ nÊTJNB EF B Z MP EFOPUBNPT QPS n a P UBNCJÊO QPS a1/n . 5FOFNPT FOUPODFT RVF QBSB a ∈ R QPTJUJWP √ 1 n a = TVQ{x ∈ R+ : xn < a} = a n = b

⇐⇒

bn = a.

4J FO FM UFPSFNB BOUFSJPS√n = a = 2 UFOFNPT RVF FYJTUF VO ÙOJDP SFBM QPTJUJWP b EFOPUBEP QPS 2, UBM RVF b2 = 2 OÙNFSP RVF OP FT SBDJPOBM



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-VFHP FYJTUFO SFBMFT RVF OP TPO SBDJPOBMFT UBMFT OÙNFSPT TPO MMBNBEPT JSSBDJPOBMFT Z EFOPUBNPT FM DPOKVOUP RVF MPT SFÙOF QPS I FT EFDJS I := R  Q. &KFSDJDJP  4FBO y ∈ I, r ∈ Q 1SVFCF RVF r + y ∈ I. 4J r = 0 FOUPODFT ry ∈ I. 5FPSFNB  4FBO a Z b SFBMFT DPO a < b &OUPODFT FYJTUFO VO SBDJPOBM r VO JSSBDJPOBM y UBMFT RVF a < r < y < b. %FNPTUSBDJÓO 'JKFNPT MPT SFBMFT a Z b 1SPCFNPT QSJNFSP FM UFPSFNB TVQPOJFOEP RVF 0 < a < b Z MVFHP BQPZBEPT FO FTUB TJUVBDJÓO QBSUJDV MBS EFNPTUSBNPT FM DBTP HFOFSBM $PNP b − a > 0 FYJTUF VO OBUVSBM N UBM RVF 1/N < b−a. %F OVFWP QPS MB QSPQJFEBE BSRVJNFEJBOB TF EFEV DF MB FYJTUFODJB EF VO OBUVSBM K UBM RVF K/N > a. -VFHP FM DPOKVOUP A := {n ∈ N : n > N a} FT OP WBDÎP Z QPS FM 1#0 FYJTUF m = NÎO A. &T DMBSP RVF m ≥ 1 Z FT FM NFOPS OBUVSBM FO A DPO MB QSPQJFEBE m/N > a. -VFHP (m − 1)/N ≤ a Z DPO FTP m m−1 1 1 = + ≤a+ < b. N N N N $PO r = m/N RVFEB QSPCBEB MB FYJTUFODJB EF VO SBDJPOBM FOUSF EPT SFBMFT QPTJUJWPT "QMJDBOEP FTUF SFTVMUBEP DPO r Z b PCUFOFNPT PUSP SB DJPOBM s UBM RVF a < r < s < b. 1PS PUSB QBSUF a<

{x ∈ R+ : x2 < 2} ⊂ {x ∈ R+ : x2 < 4} √ √ JNQMJDB RVF 0 < 2 < 2 Z BTÎ 0 < 2/2 < 1 RVF BEFNÃT FT JSSBDJPOBM -VFHP √ 2 (s − r) < s < b, a 1 EFàOBNPT FM DPOKVOUP Br,x := {ar : r ∈ Q, r ≤ x}. &TUF DPOKVOUP FT OP WBDÎP TJ x ≥ 0 FOUPODFT 1 ∈ Br,x QVFT 1 = a0 0 ∈ Q Z 0 ≤ x TJ x < 0 FYJTUF VO OBUVSBM Nx UBM RVF −Nx < x Z BTÎ a−Nx ∈ Br,x . &M DPOKVOUP Br,x SFTVMUB FTUBS BDPUBEP TVQFSJPSNFOUF FO R TFB ar ∈ Br,x . &YJTUF VO OBUVSBM Mx UBM RVF x < Mx Z QPS UBOUP r < Mx QBSB UPEP r ∈ Q DPO r ≤ x Z BTÎ aMx = aMx −r ar > ar , MP



-PT OÙNFSPT SFBMFT

RVF EFNVFTUSB RVF Br,x FTUÃ BDPUBEP TVQFSJPSNFOUF 1PS FM BYJPNB EF DPNQMFUF[ UJFOF TFOUJEP EFàOJS QBSB a > 1 Z x SFBM ax := TVQ Br,x = TVQ{ar : r ∈ Q, r ≤ x}. &O FM DBTP 0 < a < 1 EFàOJNPT ax := 1/(a−1 )x . 0CTFSWFNPT RVF DPNP ar > 0 QBSB UPEP SBDJPOBM r FOUPODFT ax > 0 QBSB UPEP x ∈ R. 0CTFSWBDJÓO /PUFNPT RVF TJ àKBNPT t ∈ Q FOUPODFT at = TVQ Br,t . &GFDUJWBNFOUF QBSB r SBDJPOBM DPO r ≤ t TF UJFOF RVF ar ≤ at MP RVF QFSNJUF DPODMVJS RVF TVQ Br,t ≤ at  1PS PUSB QBSUF DPNP at ∈ Br,t FOUPODFT at ≤ TVQ Br,t . 1SPQPTJDJÓO  4FBO a Z b SFBMFT QPTJUJWPT EJGFSFOUFT EF  DPO ab = 1 Z TFB x ∈ R &OUPODFT (ab)x = ax bx . %FNPTUSBDJÓO 'JKFNPT x ∈ R $POTJEFSFNPT QSJNFSP FM DBTP FO FM RVF a > 1 Z b > 1 &OUPODFT UFOJFOEP FO DVFOUB RVF {ar br : r ∈ Q, r ≤ x} ⊆ {ar bs : r, s ∈ Q; r, s ≤ x}, TF UJFOF RVF (ab)x = TVQ{(ab)r : r ∈ Q, r ≤ x} = TVQ{ar br : r ∈ Q, r ≤ x} ≤ TVQ{ar bs : r, s ∈ Q, r, s ≤ x} = (TVQ{ar : r ∈ Q, r ≤ x}) · (TVQ{br : r ∈ Q, r ≤ x}) = a x bx . &O MB QFOÙMUJNB JHVBMEBE QSFWJB IFNPT VTBEP FM &KFSDJDJP  E

QÃHJOB  1SPCFNPT BIPSB MB PUSB EFTJHVBMEBE 4FB ε > 0 1PS DBSBDUFSJ[BDJÓO EF TVQSFNP FYJTUFO SBDJPOBMFT r Z s UBMFT RVF r ≤ x s ≤ x Z BEFNÃT a x − ε < a r ≤ ax

Z b x − ε < b s ≤ bx .



-VFHP DPNCJOBOEP MBT QSJNFSBT EFTJHVBMEBEFT FO 

PCUFOFNPT RVF ax bx < ε2 + ε(ar + bs ) + ar bs , Z FTUP QPS MBT TFHVOEBT EFTJHVBMEBEFT FO 

FT NFOPS P JHVBM RVF ε2 + ε(ax + bx ) + ar bs  1PS UBOUP UFOFNPT RVF ax bx < ε2 + ε(ax + bx ) + ar bs = ε(ax + bx + ε) + ar bs .





5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

"IPSB CJFO QPS USJDPUPNÎB r ≤ s P CJFO r > s 4J PDVSSF MP QSJNFSP FOUPODFT OFDFTBSJBNFOUF ar bs = ar−s (ab)s ≤ (ab)s Z BTÎ QPS 

UFOFNPT RVF ax bx < ε(ax + bx + ε) + (ab)s ≤ ε(ax + bx + ε) + (ab)x .



%F NBOFSB TJNJMBS TJ r > s FOUPODFT VTBOEP ar bs = bs−r (ab)r < (ab)r Z 

DPODMVJNPT RVF ax bx < ε(ax + bx + ε) + (ab)r ≤ ε(ax + bx + ε) + (ab)x .



%F 

 Z DPNP ε > 0 FT BSCJUSBSJP TF UJFOF RVF ax bx ≤ (ab)x DPO MP DVBM RVFEB EFNPTUSBEB MB JHVBMEBE ax bx = (ab)x FO FM DBTP a > 1 Z b > 1. $POTJEFSFNPT BIPSB RVF 0 < a < 1 Z 0 < b < 1 &O FTUF DBTP 0 < ab < 1 Z UBOUP a−1 DPNP b−1 TPO NBZPSFT RVF VOP QPS UBOUP BQMJDBNPT MP BOUFSJPS QBSB PCUFOFS RVF (a−1 b−1 )x = (a−1 )x (b−1 )x  &M BSHVNFOUP TF DPNQMFUB VTBOEP MB EFàOJDJÓO EF MB TJHVJFOUF NBOFSB 1 1 1 = −1 −1 x = −1 x −1 x [(ab)−1 ]x [a b ] (a ) (b ) 1 1 = −1 x · −1 x = ax bx . (a ) (b )

(ab)x =

"IPSB TVQPOHBNPT RVF 0 < a < 1 Z b > 1. &OUPODFT bx = (baa−1 )x = (ba)x (a−1 )x 1 = (ba)x x a

TJ

ab > 1

ZB RVF

ax =

1 (a−1 )x

).

%F BRVÎ TF DPODMVZF MB JHVBMEBE EFTFBEB (ba)x = bx ax . "IPSB CJFO TJ ab < 1 FOUPODFT ax = (abb−1 )x = (ab)x (b−1 )x ZB RVF FM SFTVMUBEP WBMF TJ ab < 1 Z b−1 < 1) 1 = (ab)x x QPS EFàOJDJÓO FO FM DBTP b−1 < 1). b -B DPODMVTJÓO TF TJHVF EF FTUB JHVBMEBE Z BTÎ RVFEB EFNPTUSBEP FM UFP  SFNB  $PSPMBSJP  4FBO a Z b SFBMFT QPTJUJWPT DPO a = b, a = 1 Z b = 1. &OUPODFT (a/b)x = ax /bx .



-PT OÙNFSPT SFBMFT

%FNPTUSBDJÓO #BTUB OPUBS RVF ax = ( ab b)x = ( ab )x bx EF MP DVBM TF TJHVF  MB BTFWFSBDJÓO  1SPQPTJDJÓO  4FB a > 0, a = 1. &OUPODFT QBSB x, y SFBMFT ax+y = ax ay . %FNPTUSBDJÓO 4FBO x, y SFBMFT Z s, t SBDJPOBMFT DPO s ≤ x Z t ≤ y. -VFHP s + t ≤ x + y Z EF FTUB NBOFSB as at = as+t ≤ ax+y . 5PNBOEP FM TVQSFNP TPCSF UPEPT MPT SBDJPOBMFT s Z t DPO s ≤ x Z t ≤ y FODPOUSBNPT RVF ax ay ≤ ax+y . 1SPCFNPT BIPSB MB EFTJHVBMEBE JOWFSTB TFB ε > 0. &YJTUF VO SBDJPOBM r < x + y UBM RVF ax+y − ε < ar . -VFHP TJ δ := x + y − r > 0, FYJTUFO SBDJPOBMFT s Z t DPO x − δ/2 < s < x

Z y − δ/2 < t < y,

MP DVBM OPT QSPEVDF x + y − δ = r < s + t < x + y Z FO DPOTFDVFODJB ax ay ≥ as at = as+t > ar > ax+y − ε, FT EFDJS ax+y < ax ay + ε Z DPNP ε > 0 FT BSCJUSBSJP FOUPODFT ax+y ≤ ax ay . 2VFEB BTÎ QSPCBEB MB QSPQPTJDJÓO

 

)BTUB BIPSB IFNPT FTUVEJBEP Z QSFTFOUBEP QSPQJFEBEFT EF ax DPO a > 0 Z a = 1 " DPOUJOVBDJÓO NPTUSBNPT FOUSF PUSBT DPTBT RVF 1x = 1 QBSB UPEP SFBM x FT QPS FTUP RVF FO MB QSFTFOUBDJÓO EF MB GVO DJÓO FYQPOFODJBM x #→ ax TF FYDMVZF a = 1 QBSB OP DPOTJEFSBS FM DBTP USJWJBM 1x  $PSPMBSJP  4FB a > 0, a =  1. &OUPODFT QBSB UPEP QBS EF SFBMFT ax x, y TF UJFOF RVF y = ax−y  5BNCJÊO TF UJFOF RVF (a−1 )x = a−x Z a QBSB DVBMRVJFS SFBM x, 1x = 1. %FNPTUSBDJÓO -B QSJNFSB BàSNBDJÓO TF TJHVF EF MB JHVBMEBE ax = ax−y+y = ax−y ay . 1BSB MB TFHVOEB OPUFNPT JOJDJBMNFOUF RVF QBSB UPEP SFBM x 1 = a0 = ax−x = ax a−x ,



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

EF MP RVF TF EFTQSFOEF MB SFMBDJÓO a−x =

1 ax

∀ x ∈ R.



4VQPOHBNPT RVF a < 1 &OUPODFT QPS EFàOJDJÓO Z MVFHP VTBOEP 

DBNCJBOEP a QPS a−1 UFOFNPT RVF a−x =

1 (a−1 )−x

= (a−1 )x .

"IPSB TJ a > 1 FOUPODFT a−1 < 1 Z BTÎ QPS EFàOJDJÓO Z QPS 

(a−1 )x = 1/ax = a−x  $PO FTUP RVFEB QSPCBEB MB TFHVOEB BàSNBDJÓO  'JOBMNFOUF 1x = (aa−1 )x = ax a−x = a0 = 1.  7FSFNPT B DPOUJOVBDJÓO RVF MB GVODJÓO FYQPOFODJBM x #→ ax FT NP OÓUPOB FTUSJDUB 1SPQPTJDJÓO  4FB a > 0, a = 1 4J x < y TPO OÙNFSPT SFBMFT FOUPODFT J 4J a > 1 TF UJFOF RVF ax < ay NPOÓUPOB DSFDJFOUF FTUSJDUB  JJ 4J a < 1 TF UJFOF RVF ax > ay NPOÓUPOB EFDSFDJFOUF FTUSJDUB  %FNPTUSBDJÓO J 'JKFNPT x, y ∈ R. $PNP y − x > 0 FYJTUF N ∈ N UBM RVF 1/N < y − x Z FO DPOTFDVFODJB QPS EFàOJDJÓO ay−x ≥ a1/N > 1 -VFHP ay = ay−x ax > ax . JJ &O FTUF DBTP UFOFNPT RVF a−1 > 1 Z QPS FM SB[POBNJFOUP IFDIP FO FM MJUFSBM QSFWJP TF DVNQMF RVF (a−1 )y−x > 1 -VFHP MB EFàOJDJÓO JNQMJDB RVF ay−x = (a−11)y−x < 1 Z DPO FTUP MMFHBNPT B MP EFTFBEP FTUP  FT ay = ay−x ax < ax .  &M TJHVJFOUF SFTVMUBEP NVFTUSB RVF MB GVODJÓO FYQPOFODJBM x #→ ax FT TPCSFZFDUJWB 5FPSFNB  4FBO y > 0 Z a > 0, a = 1 SFBMFT àKPT &OUPODFT FYJTUF VO ÙOJDP SFBM z UBM RVF az = y. %FNPTUSBDJÓO 7BNPT B VTBS DPNP BOUFT MB DPNQMFUF[ EF R QBSB EF NPTUSBS MB FYJTUFODJB $POTJEFSFNPT QSJNFSP FM DBTP a > 1. 4F EFàOF FM DPOKVOUP A := {x ∈ R : ax < y}. -B BSRVJNFEJBOB HBSBOUJ[B RVF A FT OP WBDÎP QVFTUP RVF FYJTUF N OB UVSBM DPO 1/N < y(a − 1) Z BEFNÃT QPS MB EFTJHVBMEBE EF #FSOPVMMJ  N 1 aN = 1 + (a − 1) ≥ 1 + N (a − 1) > N (a − 1) > ; y



-PT OÙNFSPT SFBMFT

QPS UBOUP a−N < y FT EFDJS −N ∈ A. 7FBNPT BIPSB RVF A FT BDPUBEP TVQFSJPSNFOUF TJ x ∈ A FOUPODFT ax < y Z EF OVFWP MB BSRVJNFEJBOB OPT QSPQPSDJPOB VO OBUVSBM M RVF OP EFQFOEF EF x UBM RVF y/(a − 1) < M. &O DPOTFDVFODJB  M ≥ 1 + M (a − 1) > M (a − 1) > y aM = 1 + (a − 1) Z DPO FTP ax < aM EF MP DVBM TF EFEVDF RVF x < M QBSB UPEP x ∈ A. 4FB z := TVQ A QSPCFNPT RVF az = y 1BSB FTUP WBNPT B EFTDBSUBS MBT EFTJHVBMEBEFT az < y Z az > y 4J PDVSSJFSB MB QSJNFSB EF FMMBT FOUPODFT ya−z > 1 Z FYJTUJSÎB VO OBUVSBM n UBM RVF ya−z − 1 1 < · n a−1 $PNP BOUFT a = (a1/n )n = [1 + (a1/n − 1)]n ≥ 1 + n(a1/n − 1), a−1 < ya−z − 1 Z BTÎ az+1/n < y n MP RVF QFSNJUJSÎB EFDJS RVF z + 1/n ∈ A DPOUSBEJDJFOEP RVF z FT FM TVQSFNP EF A %F NBOFSB BOÃMPHB TJ GVFSB az > y FODPOUSBSÎBNPT VO OBUVSBM k DPO az /y − 1 1 < ; k a−1 Z SB[POBOEP WÎB MB EFTJHVBMEBE EF #FSOPVMMJ DPNP TF IJ[P PCUFOFNPT RVF (a − 1)/k ≥ a1/k − 1 EF EPOEF PCUFOESÎBNPT a1/k − 1 < az /y − 1 FT EFDJS az−1/k > y 1PS PUSB QBSUF TFSÎB QPTJCMF IBMMBS VO t ∈ A UBM RVF z − 1/k < t Z DPO FTP az−1/k < at < y, VOB DPOUSBEJDDJÓO -B VOJDJEBE TF EFTQSFOEF EF RVF az = aw ⇒ z = w. $POTJEFSFNPT BIPSB FM DBTP a < 1. &OUPODFT a−1 > 1 Z QPS MP EFNPTUSBEP QSFWJBNFOUF FYJTUF VO ÙOJDP w ∈ R UBM RVF (a−1 )w = y, FT  EFDJS a−w = y.  MP RVF JNQMJDBSÎB RVF a1/n − 1 ≤

$PNP MB GVODJÓO FYQPOFODJBM FT CJZFDUJWB FYJTUF TV JOWFSTB " DPO UJOVBDJÓO MB JOUSPEVDJNPT %FàOJDJÓO  &M OÙNFSP SFBM z UBM RVF az = y TF MMBNB FM MPHBSJUNP EF y > 0 FO CBTF a Z TF EFOPUB QPS MPHa y &O PUSPT UÊSNJOPT QBSB y > 0 az = y :⇐⇒ z = MPHa y.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&T TJNQMF WFS RVF TJ 0 < x < 1 FOUPODFT MPHa x < 0, TJ x > 1 Z MPHa x > 0 BEFNÃT MPHa (xy) = MPHa x + MPHa y. 1PS UBOUP MPHa y + MPHa y −1 = MPHa yy −1 = MPHa 1 = 0 JNQMJDB RVF MPHa y −1 = − MPHa y; Z EF FTUBT EPT QSPQJFEBEFT TF JOàFSF RVF MPHa (x/y) = MPHa x − MPHa y. -B TJHVJFOUF QSPQPTJDJÓO BEFNÃT EF TV WBMPS FO TÎ NJTNB TFSÃ EF VUJMJ EBE QBSB EFNPTUSBS MB QSPQJFEBE (ax )y = axy QBSB SFBMFT x, y. 1SPQPTJDJÓO  4FB a VO SFBM QPTJUJWP 4J {xn } FT VOB TVDFTJÓO EF SFBMFT UBM RVF xn → 0, FOUPODFT axn → 1. %FNPTUSBDJÓO &M DBTP QBSUJDVMBS a = 1 FT JONFEJBUP 4VQPOHBNPT FO UPODFT RVF a = 1. 4FBO ε > 0 Z yn := axn . -VFHP xn = MPHa yn Z QPS MB EFàOJDJÓO EF DPOWFSHFODJB UFOFNPT MB FYJTUFODJB EF VO OBUVSBM N UBM RVF − MPHa (ε + 1) < MPHa yn < MPHa (ε + 1) QBSB DBEB n ≥ N MP DVBM FT FRVJWBMFOUF B 1 < yn < ε + 1, ε+1 QBSB n ≥ N  Z EF FTUP TF EFTQSFOEF RVF −ε < −

ε < yn − 1 < ε, ε+1

TJ

n ≥ N,  

RVFEBOEP QSPCBEB MB QSPQPTJDJÓO 0CTFSWFNPT RVF TJ xn → x FOUPODFT axn = axn −x ax → ax .

1SPQPTJDJÓO  4FBO x, y SFBMFT Z a ∈ R+ àKP &OUPODFT (ax )y = axy . %FNPTUSBDJÓO &T DMBSP RVF QBSB y = 0 MB JHVBMEBE FT WÃMJEB Z FT GÃDJM QSPCBS QPS JOEVDDJÓO RVF (ax )n = axn QBSB UPEP OBUVSBM n 5BNCJÊO QBSB n OBUVSBM (ax )−n = (ax )n(−1) = [(ax )n ]−1 = (anx )−1 = =



a0 = a0−nx = a−xn , anx

1 anx

-PT OÙNFSPT SFBMFT

RVFEBOEP EFNPTUSBEP RVF (ax )m = axm QBSB UPEP m ∈ Z Z QBSB UPEP SFBM x "QMJDBOEP FTUF SFTVMUBEP UFOFNPT ax = (ax/n )n ⇒ (ax )1/n = ax/n ; Z EF FTUB NBOFSB TJ r = m/n FT VO SBDJPOBM DPO n > 0 % &n [(ax )r ]n = (ax )m/n = (ax )m = axm ⇒ (axm )1/n = (ax )r . 1FSP (axm )1/n = axm/n MP RVF EFNVFTUSB MB JHVBMEBE (ax )r = axr QBSB UPEP SBDJPOBM r 'JOBMNFOUF VTBOEP MB QSPQPTJDJÓO BOUFSJPS QSPCBNPT MB JHVBMEBE QBSB y SFBM BTÎ FYJTUF VOB TVDFTJÓO EF SBDJPOBMFT {rn } UBM RVF rn → y Z QPS UBOUP xrn → xy Z axrn → axy  -VFHP (ax )y = MÎNn→∞ (ax )rn = MÎNn→∞ axrn = axy .  

$PO FTUP IFNPT QSPCBEP MB QSPQPTJDJÓO

%F FTUB QSPQJFEBE TF EFTQSFOEF QBSB x > 0, RVF MPHa xy = y MPHa x. 1SFTFOUBNPT BIPSB VO DBTP QBSUJDVMBS JNQPSUBOUF EF EJWFSHFODJB EF VOB TVDFTJÓO %FàOJDJÓO  4FB {an } VOB TVDFTJÓO EF OÙNFSPT SFBMFT %FDJNPT RVF an → +∞ TJ EBEP DVBMRVJFS SFBM M > 0 FYJTUF VO OBUVSBM N UBM RVF n ≥ N JNQMJDB an ≥ M. %FDJNPT RVF an → −∞ TJ −an → +∞ &KFSDJDJP  4VQPOHB RVF an −→ +∞ Z bn −→ b DPO b = 0. %F NVFTUSF RVF an bn −→ +∞ TJ b > 0 Z an bn −→ −∞ TJ b < 0 y2VÊ QVFEF EFDJS EF an + bn &KFNQMP  4FBO a > 0, a = 1 Z {bn } VOB TVDFTJÓO EF SFBMFT UBM RVF bn → +∞ DVBOEP n → ∞. 1SPCFNPT RVF B 4J a > 1 FOUPODFT abn → +∞ DVBOEP n → ∞. C 4J a < 1 FOUPODFT abn → 0 DVBOEP n → ∞. B 4FB M > 0 1PS MB IJQÓUFTJT FYJTUF VO OBUVSBM N0 UBM RVF TJ n ≥ N0 M FOUPODFT bn > 1 + a−1  $PNP QBSB MPT OBUVSBMFT n ≥ N0 , bn > 1 FOUPODFT [[ bn ]] FT VO OBUVSBM NBZPS P JHVBM RVF VOP "EFNÃT [[ bn ]] + 1 > bn ≥ [[ bn ]] Z MB EFTJHVBMEBE EF #FSOPVMMJ JNQMJDBO QBSB UPEP n ≥ N0 abn ≥ a[[ bn ]] = (1 + a − 1)[[ bn ]] ≥ 1 + [[ bn ]](a − 1) > [[ bn ]](a − 1) ≥ (bn − 1)(a − 1) > M,



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

DPO MP RVF TF QSVFCB FM MJUFSBM C 1PS MB EFàOJDJÓO EF FYQPOFODJBM DPO CBTF NFOPS RVF VOP UFOFNPT RVF 1 abn = −1 bn −→ 0, DVBOEP n → ∞, (a ) ZB RVF a−1 > 1 Z QPS MP EFNPTUSBEP QSFWJBNFOUF &KFSDJDJP  4FB x ∈ R DPO x = 0 Z x < 1. %FNVFTUSF RVF QBSB UPEP OBUVSBM n ≥ 2 TF WFSJàDB (1 − x)n > 1 − nx. &KFNQMP  4FB {an }∞ n=1 MB TVDFTJÓO EF OÙNFSPT SFBMFT EFàOJEB QPS   1 n an = 1 + . n 7FBNPT RVF FTUB TVDFTJÓO FT DSFDJFOUF Z BDPUBEB Z QPS FOEF FT DPOWFS HFOUF 1SJNFSP OPUFNPT RVF QPS FM 5FPSFNB EFM CJOPNJP WFS "

QÃH 

n   n   n! n 1 an = = 2 + · (n − k)! k! nk k nk k=0

k=2

"IPSB FT GÃDJM QSPCBS QPS JOEVDDJÓO RVF QBSB DBEB OBUVSBM k k! ≥ 2k−1 Z n! ∀k ∈ N, k ≤ n TF UJFOF RVF ≤ 1. (n − k)! nk -VFHP n n   1 1 1 ≤2+ = 2 + 1 − n−1 < 3, 2 ≤ an ≤ 2 + k−1 k! 2 2 k=2

k=2

MP RVF JNQMJDB RVF 2 ≤ an < 3 QBSB DBEB OBUVSBM n Z QPS UBOUP MB TVDFTJÓO FT BDPUBEB "IPSB CJFO     n   n+1 n+1 an+1 n + 2 n+1 n + 2 n+1 n n = = an n+1 n+1 n+1 n+1 n n+1    1 n+1 n+1 1 > 1− = 1, = 1− 2 (n + 1) n n+1 n EPOEF MB ÙMUJNB EFTJHVBMEBE FT PCUFOJEB QPS MB EFTJHVBMEBE EF #FSOPVMMJ &TUP QSVFCB RVF an < an+1 QPS UBOUP MB TVDFTJÓO FT DSFDJFOUF FTUSJDUB /ÓUFTF RVF BM BQMJDBS #FSOPVMMJ MB EFTJHVBMEBE SFTVMUB FTUSJDUB QVFTUP RVF n + 1 ≥ 2 Z (n + 1)−2 < 1 WFS &KFSDJDJP  



-PT OÙNFSPT SFBMFT

0CTFSWBDJÓO $PNP FM MÎNJUF EF MB TVDFTJÓO BOUFSJPS FYJTUF Z FO SFBMJ EBE FT EF HSBO JNQPSUBODJB WBNPT B SFTBMUBSMP DPO FM OPNCSF EF OÙNFSP e   1 n . e := MÎNn→∞ 1 + n "EFNÃT DPNP e ≥ 2 QVFEF VTBSTF DPNP CBTF EF VO MPHBSJUNP FO SFBMJ EBE FTUF MPHBSJUNP TF MMBNB MPHBSJUNP OBUVSBM Z TF BDPTUVNCSB VTBS MB OPUBDJÓO QBSB x > 0, MO x. 5FPSFNB  4FB {an } VOB TVDFTJÓO EF OÙNFSPT SFBMFT OP OVMPT UBM RVF    an+1   < 1. l := MÎNn→∞  an  &OUPODFT {an } DPOWFSHF B DFSP %FNPTUSBDJÓO 6TBOEP MB EFàOJDJÓO EF DPOWFSHFODJB DPO ε = 12 (1 − l) FODPOUSBNPT VO OBUVSBM N UBM RVF QBSB n ≥ N,    an+1  1    an  < l + ε = 2 (1 + l) := c < 1. -VFHP |an+1 | < c|an | QBSB UPEP n ≥ N EF EPOEF TF EFEVDF RVF |am | < cm |aN |c−N , QBSB UPEP OBUVSBM m > N. $PNP 0 < c < 1 FOUPODFT cm −→ 0 Z FO   DPOTFDVFODJB am −→ 0 DVBOEP m −→ ∞. &KFNQMP  -BT TVDFTJPOFT

 (−1)n n! ∞ nn n=1

Z

n2 n!

∞ n=1

DPOWFSHFO B DFSP QVFT TJ an EFOPUB FM SFTQFDUJWP UÊSNJOP HFOFSBM   n  an+1    = n! (n + 1) n =  1 n −→ 1/e < 1,  an  (n + 1)n+1 n! 1 + n1 Z QBSB MB PUSB TVDFTJÓO    an+1  (n + 1)2 n! 1 1    an  = n! (n + 1) n2 = n + n2 −→ 0 < 1. 

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP  4FB a > 1 -B EFTJHVBMEBE an > n OP FT DJFSUB QBSB UPEP OBUVSBM n Z QBSB UPEP a > 1 DPNP TF QVFEF WFSJàDBS QPS FKFNQMP DPO a = 1, 01 1FSP TÎ TF DVNQMF B QBSUJS EF DJFSUP N ∈ N. -B JEFB FO FTUF FKFNQMP FT NPTUSBS RVF DJFSUPT MÎNJUFT TPO ÙUJMFT QBSB EFNPTUSBS DJFSUBT EFTJHVBMEBEFT 4J IBDFNPT bn = n/an FOUPODFT bn+1 1 1n+1 −→ < 1. = bn a n a 1PS FM 5FPSFNB  DPODMVJNPT RVF bn −→ 0 Z QPS EFàOJDJÓO EF DPO WFSHFODJB FYJTUF VO OBUVSBM N UBM RVF TJ n ≥ N FOUPODFT bn < 1 FT EFDJS an > n QBSB UPEP n ≥ N. &KFSDJDJP   4VQPOHB RVF |a| < 1 %FNVFTUSF RVF MB TVDFTJÓO {nan }∞ n=1 FT UBM RVF nan −→ 0 DVBOEP n → ∞.    n n n!  %FUFSNJOF TJ MBT TVDFTJPOFT (−1) Z 5n! DPOWFSHFO Z FO n 10 DVFOUSF TV SFTQFDUJWP MÎNJUF FO FM DBTP EF RVF FYJTUB &KFSDJDJP  4FB A ⊂ R BDPUBEP Z UBM RVF ÎOG A > 0. 4J EFàOJNPT FM DPOKVOUP EF MPT SFDÎQSPDPT EF A DPNP A1 := {1/a : a ∈ A} QSVFCF RVF FTUF DPOKVOUP FT BDPUBEP TVQFSJPSNFOUF Z BEFNÃT   1 1 = · TVQ A ÎOG A

 4VCTVDFTJPOFT Z FM UFPSFNB EF #PM[BOP8FJFSTUSBTT &M DPODFQUP EF TVCTVDFTJÓO DPNP TV OPNCSF MP TVHJFSF FT VO TVCDPO KVOUP EF VOB TVDFTJÓO EBEB QFSP FTUB BàSNBDJÓO OP FT FYBDUB &O FTUB TFDDJÓO QSFDJTBSFNPT FM DPODFQUP Z NPTUSBSFNPT TV JNQPSUBOUF QBQFM FO FM BOÃMJTJT NBUFNÃUJDP %FàOJDJÓO  4FBO {xn } VOB TVDFTJÓO EF OÙNFSPT SFBMFT Z {nk }k∈N VOB TVDFTJÓO EF OBUVSBMFT FTUSJDUBNFOUF DSFDJFOUF -B TVDFTJÓO {xnk } TF MMBNB VOB TVCTVDFTJÓO EF {xn }. &T DMBSP EF MB EFàOJDJÓO RVF UPEB TVCTVDFTJÓO FT TVCDPOKVOUP EF MB TVDFTJÓO %FTDSJCJNPT DÓNP PCUFOFS TVCTVDFTJPOFT EF VOB TVDFTJÓO EBEB VOB TVDFTJÓO {xn } EF FMMB FYUSBFNPT UÊSNJOPT DVZPT ÎOEJDFT GPSNBO VOB TF DVFODJB FTUSJDUBNFOUF DSFDJFOUF MPT UÊSNJOPT BTÎ PCUFOJEPT DPOTUJUVZFO VOB TVCTVDFTJÓO EF {xn }.



-PT OÙNFSPT SFBMFT

0USB GPSNB EF FOUFOEFS FM DPODFQUP EF TVCTVDFTJÓO FT MB TJHVJFO UF BTPDJBEB B {xn } FYJTUF VOB GVODJÓO f : N −→ R DVZP SBOHP FT QSFDJTBNFOUF MB TVDFTJÓO EF NBOFSB TJNJMBS TJ S : N −→ N FT MB GVODJÓO FTUSJDUBNFOUF DSFDJFOUF RVF EFTDSJCF MB TVDFTJÓO {nk }, FOUPO DFT MB DPNQPTJDJÓO f ◦ S : N −→ R FTUÃ CJFO EFàOJEB FT UBM RVF (f ◦ S)(k) = f (nk ) = xnk Z TV SBOHP FT QSFDJTBNFOUF MB TVCTVDFTJÓO EFUFSNJOBEB QPS S. &T EFDJS VOB TVCTVDFTJÓO TF PCUJFOF BM DPNQPOFS VOB GVODJÓO FTUSJDUBNFOUF DSFDJFOUF EF N FO N DPO MB GVODJÓO RVF EFT DSJCF MB TVDFTJÓO &O QBSUJDVMBS DPNP MB GVODJÓO JEFOUJEBE i(n) = n, FT DSFDJFOUF FTUSJDUB UPEB TVDFTJÓO FT TVCTVDFTJÓO EF TÎ NJTNB &KFNQMP  $POTJEFSFNPT MB TVDFTJÓO {1/n}∞ n=1  -B TVDFTJÓO {1/2, 1, 1/4, 1/3, 1/6, 1/5, 1/8, . . .} OP FT VOB TVCTVDFTJÓO EF {1/n} QVFT MPT UÊSNJOPT OP TPO PCUFOJEPT TJHVJFOEP VOB TFMFDDJÓO FTUSJDUBNFOUF DSFDJFO UF TF FTDPHJÓ FM TFHVOEP MVFHP FM QSJNFSP EFTQVÊT FM DVBSUP FM UFSDFSP FUD &KFSDJDJP  $POTJEFSF MB TVDFTJÓO EBEB QPS an = 1/n QBSB n JNQBS Z an = 1 TJ n FT QBS y&T FTUB TVDFTJÓO DPOWFSHFOUF &MBCPSF VO BSHVNFOUP RVF KVTUJàRVF TV SFTQVFTUB &KFNQMP  4FB {an } MB TVDFTJÓO EF SFBMFT EFàOJEB QPS a1 = 1 Z QBSB n ≥ 1, 1 an+1 = (2an + 3). 4 7FBNPT RVF FTUB TVDFTJÓO FT DPOWFSHFOUF Z IBMMFNPT TV MÎNJUF -PT QSJNFSPT DVBUSP UÊSNJOPT 1, 5/4, 11/8, 23/16 OPT JOEVDFO B QFOTBS RVF EJDIB TVDFTJÓO FT DSFDJFOUF FTUSJDUB 1SPCFNPT FTUB BàSNB DJÓO QPS JOEVDDJÓO QBSB n = 1 MB BàSNBDJÓO FT JONFEJBUB 4VQPOHBNPT RVF an < an+1 Z EFNPTUSFNPT RVF an+1 < an+2  1 1 1 an+1 − an+2 = (2an + 3) − (2an+1 + 3) = (an − an+1 ) < 0. 4 4 2 -B ÙMUJNB EFTJHVBMEBE TF EFCF B MB IJQÓUFTJT JOEVDUJWB 7FBNPT BIPSB RVF {an } FT BDPUBEB TVQFSJPSNFOUF VOB FYQMPSBDJÓO TPCSF BMHVOPT UÊS NJOPT EF ÎOEJDF NBZPS OPT QFSNJUFO JOUVJS RVF 3/2 FT VOB DPUB TVQFSJPS EF MB TVDFTJÓO MP DVBM UBNCJÊO QSPCBSFNPT QPS JOEVDDJÓO &M QBTP CBTF FT DMBSP QBSB FM QBTP JOEVDUJWP TVQPOHBNPT RVF an < 3/2 Z WFBNPT RVF FTUB EFTJHVBMEBE UBNCJÊO FT DJFSUB QBSB FM UÊSNJOP (n + 1)ÊTJNP 1PS EFàOJDJÓO UFOFNPT RVF an+1 = 14 (2an + 3), RVF QPS MB IJQÓUFTJT EF JOEVDDJÓO FT NFOPS RVF 14 (3 + 3) = 3/2 RVFEBOEP QSPCBEB BTÎ MB BTFWF SBDJÓO TPCSF FM BDPUBNJFOUP TVQFSJPS %FCJEP BM 5FPSFNB  DPODMVJNPT RVF {an } DPOWFSHF 7FBNPT BIPSB DÓNP FODPOUSBS TV MÎNJUF 6OB GPSNB



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

DPOTJTUF FO BQMJDBS FM DJUBEP UFPSFNB FM DVBM EJDF RVF FM MÎNJUF CVTDBEP FT TVQ an  &TUP SFRVJFSF DBMDVMBS FTF TVQSFNP RVF EFCF TFS NFOPS P JHVBM RVF 3/2 MP RVF QPTJCMFNFOUF OP TFB UBO TFODJMMP EF MMFWBS B DBCP &M PUSP DBNJOP NVDIP NÃT TJNQMF DPOTJTUF FO UPNBS MÎNJUF FO MB JHVBMEBE RVF EFàOF an+1 UFOJFOEP QSFTFOUF RVF MÎN an = MÎN an+1 = MÎN an+1 .

n→∞

n→∞

n→∞

4J EFOPUBNPT EJDIP MÎNJUF QPS - UFOFNPT RVF - = 14 (2- + 3) EF EPOEF TF EFEVDF RVF - = 3/2 &KFSDJDJP  4FB {xn } MB TVDFTJÓO FO R EBEB QPS x1 > 1 Z QBSB n ≥ 1, xn+1 = 2−1/xn  1SVFCF RVF MB TVDFTJÓO FT NPOÓUPOB Z BDPUBEB &ODVFOUSF TV MÎNJUF &KFSDJDJP  4FB an ≥ 0 QBSB UPEP OBUVSBM n. 4J an −→ a DVBOEP √ √ n → ∞, QSVFCF RVF a ≥ 0 Z BEFNÃT RVF an −→ a. &KFSDJDJP  "SHVNFOUF MB DPOWFSHFODJB P OP EF MB TVDFTJÓO EFàOJEB QPS J a1 = 1 Z QBSB n ≥ 1, an+1 = 12 an + 2. JJ x1 > 0, xn+1 = 2 +

1 xn

QBSB n ≥ 1.

JJJ y1 = 1 Z QBSB n ≥ 1, yn+1 =

√

2 + yn .

&ODVFOUSF FM MÎNJUF EF MBT RVF TFBO DPOWFSHFOUFT &KFSDJDJP  4FB {xn } MB TVDFTJÓO EF SFBMFT EBEB QPS x1 > 0, xn+1 = xn +

1 QBSB n ≥ 1. x2n

1SVFCF RVF {xn } OP FT BDPUBEB &KFSDJDJP  4J {an } FT VOB TVDFTJÓO DPOWFSHFOUF Z TVT UÊSNJOPT TPO BMUFSOBUJWBNFOUF EF TJHOP QPTJUJWP OFHBUJWP QPTJUJWP w yDVÃM FT FM WBMPS EFM MÎNJUF &YQMJRVF TV SFTQVFTUB 1SPQPTJDJÓO  4FB {an } VOB TVDFTJÓO EF OÙNFSPT OBUVSBMFT UBM RVF an → +∞ &OUPODFT (1 + 1/an )an → e. %FNPTUSBDJÓO 4FB ε > 0 $PNP {(1 + 1/n)n } DPOWFSHF BM OÙNFSP e FYJTUF VO OBUVSBM N1 UBM RVF TJ n ≥ N1 FOUPODFT |(1 + 1/n)n − e| < ε/2



-PT OÙNFSPT SFBMFT

$PNP UBNCJÊO FT VOB TVDFTJÓO EF $BVDIZ FOUPODFT FYJTUF VO OBUVSBM N2 UBM RVF k, m ≥ N2 ⇒ |(1 + 1/k)k − (1 + 1/m)m | < ε/2. 1PS PUSP MBEP DPNP an → +∞ FYJTUF VO OBUVSBM N3 UBM RVF TJ n > N3 FOUPODFT an ≥ N2 . &O DPOTFDVFODJB TJ N := NÃY{N1 , N2 , N3 } Z n ≥ N FOUPODFT |(1 + 1/an )an − e| ≤ |(1 + 1/an )an − (1 + 1/n)n | + |(1 + 1/n)n − e| < ε/2 + ε/2 = ε, MP DVBM EFNVFTUSB MB QSPQPTJDJÓO  n 2 ∞ &KFNQMP  -B TVDFTJÓO 1 + 1/n2

  DPOWFSHF BM OÙNFSP e

n=1 n2  /PUFNPT

RVF FTUB TVDF #BTUB BQMJDBS FM SFTVMUBEP QSFWJP DPO an = TJÓO FT VOB TVCTVDFTJÓO EF {(1 + 1/n)n } QVFT MB GVODJÓO n #−→ n2 FT FTUSJDUBNFOUF DSFDJFOUF $PSPMBSJP  4FB {an } VOB TVDFTJÓO EF OÙNFSPT SFBMFT UBM RVF an → +∞ &OUPODFT TF DVNQMF RVF (1 + 1/an )an → e. %FNPTUSBDJÓO $PNP an → +∞ QPEFNPT TVQPOFS RVF an > 1 /PUF NPT RVF QPS EFàOJDJÓO EF QBSUF FOUFSB 1 ≤ [[ an ]]. 5BNCJÊO bn := 1 + 1/an > 1 Z BEFNÃT [[ an ]] ≤ an < [[ an ]] + 1.



%F FTUP TF EFEVDF RVF [[ an ]] → +∞ Z bn[[ an ]] ≤ bann < b[[nan ]]+1 = (1 + 1/an )[[ an ]]+1 .



0CTFSWFNPT RVF EF MB QSJNFSB EFTJHVBMEBE FO  TF EFEVDF RVF bn ≤ 1 + 1/[[ an ]] [[ a ]]+1

≤ (1 + 1/[[ an ]])[[ an ]]+1 , MP DVBM DPNCJOBEP Z BTÎ MMFHBNPT B RVF bn n DPO MB TFHVOEB EFTJHVBMEBE FO  OPT BSSPKB MB EFTJHVBMEBE bann < (1 + 1/[[ an ]])[[ an ]]+1 = (1 + 1/[[ an ]])[[ an ]] (1 + 1/[[ an ]]). 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&T JNQPSUBOUF OPUBS RVF QPS MB 1SPQPTJDJÓO  FM MBEP EFSFDIP EF  DPOWFSHF BM OÙNFSP e DVBOEP n → ∞. "IPSB CJFO MB TFHVOEB EFTJHVBMEBE FO  QSPEVDF bn > 1 +

1 , [[ an ]] + 1

Z BTÎ QPS MB QSJNFSB EFTJHVBMEBE FO 

UFOFNPT RVF bann ≥ b[[nan ]] =

[[ an ]]+1

bn

bn

>

1 1 )[[ an ]]+1 . (1 + bn [[ an ]] + 1



$PNP FM MBEP EFSFDIP EF  DPOWFSHF BM OÙNFSP e BM DPNCJOBS FTUP DPO  PCUFOFNPT FM SFTVMUBEP EFTFBEP Z QPS UBOUP MB QSVFCB RVFEB  DPNQMFUB  6O SFTVMUBEP TFODJMMP QFSP EF VUJMJEBE TF QSFTFOUB FO FM TJHVJFOUF MFNB Z TV EFNPTUSBDJÓO FKFSDJDJP TF IBDF QPS JOEVDDJÓO -FNB  4FB {xnk } VOB TVCTVDFTJÓO EF {xn } &OUPODFT QBSB UPEP OBUVSBM k TF UJFOF RVF nk ≥ k. 5FPSFNB  4FBO {xn } VOB TVDFTJÓO EF SFBMFT Z L VOB DPOTUBOUF &OUPODFT {xn } DPOWFSHF B L TJ Z TPMP TJ UPEB TVCTVDFTJÓO EF FMMB UBNCJÊO DPOWFSHF B L. %FNPTUSBDJÓO $POEJDJÓO OFDFTBSJB TFBO {xnk } VOB TVCTVDFTJÓO EF {xn } Z ε > 0. $PNP xn −→ L FYJTUF VO OBUVSBM N UBM RVF TJ n > N FOUPODFT |xn − L| < ε. 4FB k > N  QPS UBOUP UFOJFOEP FO DVFOUB RVF nk ≥ k PCUFOFNPT RVF nk > N Z BTÎ DPODMVJNPT RVF |xnk − L| < ε QBSB UPEP k > N MP RVF TJHOJàDB RVF xnk −→ L DVBOEP k → ∞. $POEJDJÓO TVàDJFOUF TVQPOHBNPT RVF MB TVDFTJÓO {xn } OP DPOWFSHF B L. -VFHP FYJTUF VO εo > 0 UBM RVF ∀m ∈ N ∃n > m

Z |xn − L| ≥ εo .



"QMJDBOEP  JOEVDUJWBNFOUF DPOTFHVJNPT VOB TVCTVDFTJÓO QBSB m = 1 FYJTUF n1 > 1 UBM RVF |xn1 − L| ≥ εo . %F OVFWP QBSB m = n1 FYJTUF n2 > n1 UBM RVF |xn2 − L| ≥ εo . 4VQPOHBNPT RVF IFNPT PCUFOJ EP OBUVSBMFT n1 < n2 < · · · < nk UBMFT RVF |xnk − L| ≥ εo . )BDJFOEP m = nk FO 

FODPOUSBNPT nk+1 > nk UBM RVF |xnk+1 − L| ≥ εo . %F FTUB NBOFSB IFNPT DPOTUSVJEP VOB TVCTVDFTJÓO EF {xn } RVF OP DPO WFSHF B L MP DVBM WB FO DPOUSBWÎB DPO MB IJQÓUFTJT &TUB DPOUSBEJDDJÓO   EFNVFTUSB RVF MB TVDFTJÓO {xn } DPOWFSHF B L.



-PT OÙNFSPT SFBMFT n

n &KFSDJDJP  %FUFSNJOF TJ MB TVDFTJÓO { (−1) 2n+1 } DPOWFSHF P OP &YQMJRVF

0CTFSWBNPT RVF FM UFPSFNB BOUFSJPS OPT QSPQPSDJPOB VO DSJUFSJP EF OP DPOWFSHFODJB EF VOB TVDFTJÓO B VO WBMPS EBEP FTUP FT {xn } OP DPO WFSHF B L TJ Z TPMP TJ FYJTUF VOB TVCTVDFTJÓO EF FMMB RVF OP DPOWFSHF B L. /ÓUFTF UBNCJÊO RVF QBSB DPODMVJS MB DPOWFSHFODJB EF VOB TVDFTJÓO B VO WBMPS EBEP FT OFDFTBSJP RVF UPEBT MBT TVCTVDFTJPOFT EF FMMB DPO WFSKBO B EJDIP WBMPS &M TJHVJFOUF SFTVMUBEP EJDF RVF DPO BM NFOPT VOB TVCTVDFTJÓO RVF DPOWFSKB TJFNQSF RVF TFB EF $BVDIZ FT TVàDJFOUF QBSB DPODMVJS MB DPOWFSHFODJB EF MB TVDFTJÓO 5FPSFNB  4FBO {xn } VOB TVDFTJÓO EF $BVDIZ Z {xnk } VOB TVCTV DFTJÓO RVF DPOWFSHF B L. &OUPODFT MB TVDFTJÓO DPOWFSHF B L. %FNPTUSBDJÓO 4FB ε > 0 $PNP {xn } FT EF $BVDIZ FYJTUF VO OBUVSBM No UBM RVF QBSB n > No Z m > No TF UJFOF RVF |xn − xm | < ε. 1PS PUSB QBSUF FYJTUF N1 ∈ N UBM RVF TJ j > N1 FOUPODFT |xnj − L| < ε. -VFHP TJ j > NÃY{No , N1 } := N UFOFNPT RVF nj ≥ j > N ≥ Ni QBSB i = 0, 1 MP DVBM JNQMJDB RVF |xnj − xj | < ε Z |xnj − L| < ε. &O DPOTFDVFODJB QBSB j > N, |xj − L| ≤ |xj − xnj | + |xnj − L| < 2ε. &TUP RVJFSF EFDJS RVF xn → L.

 

5FPSFNB  5FPSFNB EF #PM[BOP8FJFSTUSBTT  5PEB TVDFTJÓO BDPUBEB EF OÙNFSPT SFBMFT QPTFF VOB TVCTVDFTJÓO DPOWFSHFOUF %FNPTUSBDJÓO 4FB {an } VOB TVDFTJÓO BDPUBEB EF OÙNFSPT SFBMFT 7B NPT B TVQPOFS RVF MB TVDFTJÓO OP DPOTUJUVZF VO DPOKVOUP àOJUP QVFT FO FTF DBTP FM BSHVNFOUP FT TJNQMF Z TF EFEVDF TFHÙO FM DBTP QBSUJDV MBS $POTUSVJNPT DPNP FO MB QSVFCB EFM 5FPSFNB  MB TVDFTJÓO {bn } EBEB QPS bn = TVQ{an , an+1 , an+2 , . . .}, MB RVF SFTVMUB EFDSFDJFOUF Z BDPUBEB JOGFSJPSNFOUF QPS UBOUP FYJTUF VO OÙNFSP SFBM L BM DVBM DPOWFSHF %PT QSPQJFEBEFT JNQPSUBOUFT DBSBDUFSJ [BO FTUF OÙNFSP L MBT DVBMFT TFSÃO ÙUJMFT FO MP RVF SFTUB EF MB QSVFCB "àSNBDJÓO  1BSB UPEP ε > 0 FYJTUF N ∈ N UBM RVF TJ n ≥ N FOUPODFT an < L + ε. &O FGFDUP DPNP {bn } FT EFDSFDJFOUF Z L = ÎOG bn FYJTUF N ∈ N UBM RVF TJ n ≥ N FOUPODFT L + ε > bN ≥ bn ≥ an 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

"àSNBDJÓO  1BSB UPEP ε > 0 TF DVNQMF RVF (∀ n ∈ N) (∃ m ≥ n) (am > L − ε). 4J FTUB BàSNBDJÓO OP GVFTF DJFSUB FYJTUJSÎB VO ε0 > 0 UBM RVF (∃ N ∈ N) (∀ m ≥ N ) (am ≤ L − ε0 ). &TUP JNQMJDBSÎB QPS EFàOJDJÓO EF bN RVF bN ≤ L − ε0 . $PNP L − ε0 ≤ bN − ε0 TF UFOESÎB MB DPOUSBEJDDJÓO bN ≤ bN − ε0 . 1PS UBOUP MB BàSNBDJÓO  FT DJFSUB 6TBNPT MB BàSNBDJÓO  DPO FM NJTNP ε > 0 Z N ∈ N EF MB BàSNBDJÓO  Z BTÎ IBMMBNPT n1 ≥ N UBM RVF an1 > L − ε. 7PMWFNPT B BQMJDBS MB TFHVOEB BàSNBDJÓO DPO FM NJTNP ε > 0 Z FM OBUVSBM n1 + 1 QBSB PCUFOFS n2 > n1 UBM RVF an2 > L − ε. 4VQPOHBNPT PCUFOJEPT OBUVSBMFT n1 > n2 > · · · > nk UBMFT RVF anj > L − ε QBSB UPEP j = 1, 2, . . . , k. 1BSB FM QBTP JOEVDUJWP BQMJDBNPT MB BàSNBDJÓO  DPO FM OBUVSBM nk + 1 Z FM NJTNP ε > 0 EF MB QSJNFSB BàSNBDJÓO QBSB FODPOUSBS nk+1 > nk UBM RVF ank+1 > L − ε. %F FTUB NBOFSB IFNPT IBMMBEP VOB TVCTVDFTJÓO {ank } UBM RVF L − ε < ank < L + ε QBSB UPEP k ∈ N. &O DPOTFDVFODJB RVFEB QSPCBEB MB FYJTUFODJB EF VOB TVCTVDFTJÓO {ank }  RVF DPOWFSHF BM OÙNFSP SFBM L.  0CTFSWBDJÓO -B QSVFCB UBNCJÊO FT FYJUPTB TJ TF EFàOF MB TVDFTJÓO cn = ÎOG {an , an+1 , an+2 , . . .}. &TUB TVDFTJÓO SFTVMUB DSFDJFOUF Z BDPUBEB QPS UBOUP FYJTUF l ∈ R UBM RVF cn → l DVBOEP n → ∞. &O FTUF DBTP MBT DPSSFTQPOEJFOUFT BàSNBDJPOFT RVF DBSBDUFSJ[BO FM OÙNFSP SFBM l TPO MBT TJHVJFOUFT "àSNBDJÓO 1 . 1BSB UPEP ε > 0 FYJTUF N ∈ N UBM RVF TJ n ≥ N FOUPODFT an > l − ε. "àSNBDJÓO 2 . 1BSB UPEP ε > 0 TF DVNQMF RVF (∀ n ∈ N) (∃ m ≥ n) (am < l + ε). $PNCJOBOEP MB QSVFCB BOUFSJPS Z MBT EFàOJDJPOFT EBEBT FO MB PCTFSWB DJÓO EF MB QÃHJOB  QPEFNPT DPODMVJS RVF TJ L = MÎN TVQ an , n→∞



-PT OÙNFSPT SFBMFT

FOUPODFT FYJTUF VOB TVCTVDFTJÓO {ank } UBM RVF ank → L DVBOEP k → ∞. 5BNCJÊO QPS MB QSJNFSB QBSUF EF FTUB PCTFSWBDJÓO TF UJFOF RVF TJ l = MÎN JOG an , n→∞

FOUPODFT FYJTUF VOB TVCTVDFTJÓO {anj } UBM RVF anj → l DVBOEP j → ∞. &KFSDJDJP  4FB x1 > 0 Z TVQPOHBNPT RVF xn+1 = xn + 1/xn QBSB UPEP OBUVSBM n ≥ 1. %FNVFTUSF RVF xn −→ +∞ DVBOEP n → ∞. &KFSDJDJP  4FBO S VO TVCDPOKVOUP OP WBDÎP EF R Z u ∈ R. .VFTUSF RVF u = TVQ S TJ Z TPMP TJ (∀ n ∈ N∗ )(u − TVQFSJPS EF S 

1 n

OP FT VOB DPUB TVQFSJPS EF S Z u +

1 n

FT VOB DPUB

&KFSDJDJP  4FB A ⊂ (0, +∞) OP WBDÎP 1SVFCF RVF u = TVQ A TJ Z TPMP TJ i) ∀ x ∈ A x ≤ u ii) ∀ d > 1 ∃ a ∈ A : ud < a. &KFSDJDJP  %FDJNPT RVF VO QBS PSEFOBEP (A, B) EF TVCDPOKVOUPT EJTKVOUPT OP WBDÎPT EF R FT VOB DPSUBEVSB TJ A ∪ B = R Z BEFNÃT (∀ a ∈ A) (∀ b ∈ B) (a < b). 4J (A, B) FT VOB DPSUBEVSB FO R QSVFCF RVF TVQ A = ÎOG B. &KFSDJDJP  4FBO a, b ∈ R àKPT Z A := {x ∈ R : a < x < b} 1SVFCF RVF ÎOG A = a Z TVQ A = b &KFSDJDJP  4FB {an } VOB TVDFTJÓO EF SFBMFT UBM RVF a2n → 0 Z a2n+1 → 0 %FNVFTUSF RVF an → 0. n  1 &KFSDJDJP  4VQPOHB RVF MB TVDFTJÓO {bn } EBEB QPS bn = k! k=1 DPOWFSHF B MB DPOTUBOUF A ∈ R &ODVFOUSF FM MÎNJUF FO UÊSNJOPT EF MB DPOTUBOUF A EF MB TVDFTJÓO {an } EBEB QPS

an =

n  3k 2 − 4k + 2 k=1

k!

·

&KFSDJDJP  6O TVCDPOKVOUP S EF OÙNFSPT SFBMFT TF EJDF RVF FT JOEVDUJWP TJ TF DVNQMFO MBT DPOEJDJPOFT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

J 0 ∈ S J (∀ x) (x ∈ S ⇒ x + 1 ∈ S). %FNVFTUSF MBT BàSNBDJPOFT B &M DPOKVOUP EF MPT OÙNFSPT OBUVSBMFT FT VO DPOKVOUP JOEVDUJWP C -B JOUFSTFDDJÓO EF UPEPT MPT TVCDPOKVOUPT JOEVDUJWPT FT VO DPOKVOUP JOEVDUJWP ' D N = {I ⊆ R : I FT VO DPOKVOUP JOEVDUJWP}. E 4J A ⊆ N Z A FT VO DPOKVOUP JOEVDUJWP FOUPODFT A = N.

 5PQPMPHÎB EF MB SFDUB SFBM )BTUB BIPSB BMHVOPT EF MPT DPODFQUPT P SFTVMUBEPT QSFTFOUBEPT IBO JOWP MVDSBEP JOUFSWBMPT BCJFSUPT P DFSSBEPT &TUF UJQP EF JOUFSWBMPT TPO DBTPT QBSUJDVMBSFT EF VOB DMBTF NÃT BNQMJB EF DPOKVOUPT MMBNBEPT BCJFSUPT P DFSSBEPT SFTQFDUJWBNFOUF MPT DVBMFT TPO MPT FMFNFOUPT EF MP RVF TF DP OPDF DPNP 5PQPMPHÎB -B OPDJÓO EF UPQPMPHÎB OP FT JNQPSUBOUF BIPSB TJO FNCBSHP QBSB DMBSJEBE Z QBSB OP DPOGVOEJS BM MFDUPS QSJODJQJBOUF TF FOUJFOEF DPNP VOB DPMFDDJÓO EF TVCDPOKVOUPT EF VO DPOKVOUP EBEP X MMBNBEPT BCJFSUPT RVF FT DFSSBEB CBKP VOJPOFT BSCJUSBSJBT JOUFSTFDDJPOFT àOJUBT Z UBOUP X DPNP FM DPOKVOUP WBDÎP FTUÃO FO MB DPMFDDJÓO %FàOJDJÓO  4FBO a ∈ R àKP Z ε > 0. -MBNBNPT CPMB FO R EF DFOUSP a Z SBEJP ε BM JOUFSWBMP BCJFSUP (a − ε, a + ε) 0USPT OPNCSFT RVF TF TVFMFO VTBS TPO FOUPSOP P WFDJOEBE EF SBEJP ε &O DVBMRVJFSB EF FTUPT DBTPT IBCMBSFNPT EF εCPMB εFOUPSOP P εWFDJOEBE EFM QVOUP a Z MB EFOPUBSFNPT QPS #ε (a). 0CTFSWFNPT RVF #ε (a) := (a − ε, a + ε) = {x ∈ R : |x − a| < ε}, MP DVBM EFKB DMBSJEBE BDFSDB EFM OPNCSF EF CPMB &KFNQMP  $VBMRVJFS JOUFSWBMP BCJFSUP BDPUBEP (a, b) DPO a < b FT b−a VOB CPMB FO R EF DFOUSP a+b 2 Z SBEJP 2 . %FàOJDJÓO  4FBO S ⊆ R DPO S OP WBDÎP Z a ∈ R àKPT %FDJNPT RVF a FT VO QVOUP EF BDVNVMBDJÓO EF S TJ UPEB εWFDJOEBE EFM QVOUP a DPOUJFOF VO QVOUP EF S EJTUJOUP EF a &O PUSPT UÊSNJOPT TJ QBSB UPEP ε > 0 TF UJFOF RVF #ε (a) ∩ (S  {a}) = ∅. &M DPOKVOUP EF QVOUPT EF BDVNVMBDJÓO EF VO DPOKVOUP S MP EFOPUBNPT QPS S  .



-PT OÙNFSPT SFBMFT

&KFNQMP  $POTJEFSFNPT MBT TJHVJFOUFT JMVTUSBDJPOFT J 4FB S = [0, 1]. &OUPODFT 0 FT QVOUP EF BDVNVMBDJÓO EF S ZB RVF EBEP ε > 0 UFOFNPT RVF (−ε, ε) ∩ (S  {0}) = (−ε, ε) ∩ (0, 1] = ∅. JJ 4FB S = {1/n}∞ n=1 . &OUPODFT  FT QVOUP EF BDVNVMBDJÓO EF S ZB RVF QBSB DVBMRVJFS OÙNFSP ε > 0 DPNP 1/n → 0 FYJTUF VO OBUVSBM N0 UBM RVF TJ n ≥ N0 FOUPODFT 1/n < ε. 1PS UBOUP 1/n ∈ (−ε, ε) ∩ (S  {0}) QBSB UPEP n ≥ N0 . JJJ &M MJUFSBM QSFWJP TF DVNQMF FO HFOFSBM FTUP FT TJ FYJTUF VO OBUVSBM N0 UBM RVF QBSB UPEP n ≥ N0 , an = a Z an → a FOUPODFT a FT VO QVOUP EF BDVNVMBDJÓO EF MB TVDFTJÓO {an } JW 4FB S = (0, 1) ∪ {2} &M QVOUP a = 2 OP FT VO QVOUP EF BDVNVMB DJÓO EF S ZB RVF FYJTUF QPS FKFNQMP ε = 1/2 UBM RVF (2 − ε, 2 + ε) ∩ (S  {2}) = (3/2, 5/2) ∩ (0, 1) FT WBDÎP 4J EFUBMMBNPT OVFWBNFOUF MPT USFT QSJNFSPT ÎUFNT OPT EBNPT DVFOUB EF EPT DPTBT QSJNFSP OP OFDFTBSJBNFOUF VO QVOUP EF BDVNVMBDJÓO EF VO DPOKVOUP FT VO FMFNFOUP EF EJDIP DPOKVOUP Z TFHVOEP RVF UPEB ε WFDJOEBE EF VO QVOUP EF BDVNVMBDJÓO a EF VO DPOKVOUP S FO SFBMJEBE DPOUJFOF JOàOJUPT QVOUPT EF S. &TUF IFDIP MP SFTBMUBNPT FO FM TJHVJFOUF SFTVMUBEP 5FPSFNB  4FB a ∈ R VO QVOUP EF BDVNVMBDJÓO EF S &OUPODFT UPEB εWFDJOEBE EF a DPOUJFOF JOàOJUPT QVOUPT EF S %FNPTUSBDJÓO -B QSVFCB RVF QSFTFOUBNPT FT JOEJSFDUB FTUP FT TVQPO HBNPT RVF FYJTUF VOB εWFDJOEBE EF a RVF DPOUJFOF VO DPOKVOUP àOJUP EF QVOUPT EF S %JHBNPT RVF QBSB BMHÙO ε > 0, #ε (a) ∩ (S  {a}) = {x1 , x2 , . . . , xm }.



-B JEFB BIPSB FT WFS RVF FYJTUF VOB WFDJOEBE EF a RVF OP TBUJTGBDF MB EFàOJDJÓO EF QVOUP EF BDVNVMBDJÓO 5BM WFDJOEBE TF DPOTUSVZF FMJHJFOEP VO OÙNFSP QPTJUJWP NFOPS RVF MBT EJTUBODJBT FOUSF a Z MPT xj MBT DVBMFT TPO QPTJUJWBT .ÃT QSFDJTBNFOUF UPNFNPT δ := NÎO {|xj − a| : j = 1, 2, . . . , m}



7FS FM BQÊOEJDF EPOEF IBDFNPT VOB CSFWF QSFTFOUBDJÓO EFM DPODFQUP EF DPOKVOUP JOàOJUP



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

Z DPOTJEFSFNPT #δ/2 (a) "àSNBNPT RVF FTUB δ/2WFDJOEBE EF a OP DPO UJFOF QVOUPT EF S  {a}. &O FGFDUP TJ FYJTUJFSB z ∈ #δ/2 (a) ∩ (S  {a}) FOUPODFT |z − a| < δ/2 Z z ∈ S  {a} "IPSB CJFO VTBOEP  UFO ESÎBNPT RVF |z − a| < δ/2 < δ ≤ |xj − a| < ε FTUP FT |z − a| < ε Z BTÎ DPODMVJSÎBNPT RVF z = xk QBSB BMHÙO k ∈ {1, 2, . . . , m}, MP DVBM OPT   BSSPKBSÎB MB DPOUSBEJDDJÓO δ ≤ |xk − a| = |z − a| < δ/2. $PSPMBSJP  5PEP DPOKVOUP àOJUP DBSFDF EF QVOUPT EF BDVNVMBDJÓO &KFSDJDJP  4FBO A Z B TVCDPOKVOUPT EF R %FNVFTUSF B 4J A ⊆ B FOUPODFT A ⊆ B   C (A ∪ B) = A ∪ B   4VQPOHBNPT RVF a ∈ A OP FT QVOUP EF BDVNVMBDJÓO EF A &OUPODFT FYJTUF VOB δCPMB EF a EJHBNPT #δ , UBM RVF #δ ∩ A ∩ {a}c = ∅, FT EFDJS FYJTUF VOB δCPMB EF a RVF TPMP DPOUJFOF VO QVOUP EFM DPOKVOUP A &O TÎNCPMPT #δ ∩ A = {a}. &TUF UJQP EF QVOUPT UJFOF OPNCSF QSPQJP %FàOJDJÓO  4FBO A VO DPOKVOUP FO MB SFDUB SFBM Z a ∈ A %FDJNPT RVF a FT VO QVOUP BJTMBEP EF A TJ FYJTUF VOB δCPMB EF a EJHBNPT #δ , UBM RVF #δ ∩ A = {a}. %FàOJDJÓO  6O TVCDPOKVOUP G ⊆ R TF EJDF RVF FT BCJFSUP TJ QBSB DBEB x ∈ G FYJTUF VOB δWFDJOEBE EF x, #δ (x) UBM RVF #δ (x) ⊂ G. 0CTFSWBDJÓO &M OÙNFSP QPTJUJWP δ QSFTFOUBEP FO MB EFàOJDJÓO EF DPO KVOUP BCJFSUP EFQFOEF EFM QVOUP x ∈ G DPNP WFSFNPT FO QSÓYJNBT JMVTUSBDJPOFT &KFNQMP  B &M JOUFSWBMP BCJFSUP (0, 1) FT VO DPOKVOUP BCJFSUP FO R TFB x ∈ (0, 1) BSCJUSBSJP QFSP àKP 7FBNPT RVF FYJTUF VOB δWFDJOEBE EF x DPOUFOJEB FO (0, 1) &O FGFDUP UPNFNPT DPNP SBEJP EF MB CPMB B FYIJCJS FM OÙNFSP δx ≡ δ := NÎO {1−x, x} OPUF RVF FM SBEJP EF QFOEF EFM QVOUP x  "àSNBNPT RVF #δ (x) ⊂ (0, 1) TJ y ∈ #δ (x) FOUPODFT QPS MB EFàOJDJÓO EFM OÙNFSP δ > 0 UFOFNPT RVF |y − x| < δ ≤ 1 − x

Z

|y − x| < δ ≤ x,

EF EPOEF TF TJHVF RVF 0 < y < 1 FTUP FT y ∈ (0, 1).



-PT OÙNFSPT SFBMFT

C &M SB[POBNJFOUP FO FM ÎUFN QSFWJP TF QVFEF FYUFOEFS B JOUFSWBMPT HFOFSBMFT EF MB GPSNB (a, b) DPO a < b. $PODSFUBNFOUF FM JOUFSWBMP BCJFSUP (a, b) FT VO DPOKVOUP BCJFSUP FO R FO QBSUJDVMBS UPEB εCPMB FT VO DPOKVOUP BCJFSUP D &T JONFEJBUP EF MB EFàOJDJÓO RVF FM DPOKVOUP EF MPT OÙNFSPT SFBMFT FT VO DPOKVOUP BCJFSUP FO R E &M DPOKVOUP WBDÎP FT VO DPOKVOUP BCJFSUP FO R ZB RVF MB TJHVJFOUF BàSNBDJÓO FT WFSEBEFSB (∀ x ∈ ∅) (∃ δ > 0 : #δ (x) ⊂ ∅). F 4FB a ∈ R àKP &OUPODFT MPT JOUFSWBMPT OP BDPUBEPT (a, +∞) Z (−∞, a) TPO DPOKVOUPT BCJFSUPT FO R FTDPKBNPT x ∈ (a, +∞) Z TFB δx = δ = x − a > 0 &OUPODFT FT JONFEJBUB MB JODMVTJÓO #δ (x) ⊂ (a, +∞) &M SB[POBNJFOUP FT TJNJMBS QBSB (−∞, a) G 'JKFNPT a < b 1PS FM ÎUFN BOUFSJPS FM DPOKVOUP A := (−∞, a) ∪ (b, +∞) SFTVMUB TFS VO DPOKVOUP BCJFSUP FO R. H &M JOUFSWBMP S = (0, 1] OP FT VO DPOKVOUP BCJFSUP FO R &TUP TF FYQMJDB EF MB TJHVJFOUF NBOFSB FYJTUF VO QVOUP FO S FM 1 RVF OP FT FM DFOUSP EF BMHVOB CPMB MB DVBM RVFEF DPOUFOJEB FO S. 0CTFSWFNPT RVF FO FM QFOÙMUJNP FKFNQMP FM DPNQMFNFOUP EF A Ac := R  A = [a, b] FM DVBM BDPTUVNCSBNPT MMBNBS JOUFSWBMP DFSSB EP $PO FTUP JOUSPEVDJNPT FM TJHVJFOUF DPODFQUP %FàOJDJÓO  6O TVCDPOKVOUP F EF R TF EJDF RVF FT VO DPOKVOUP DFSSBEP FO R TJ TV DPNQMFNFOUP F c FT VO DPOKVOUP BCJFSUP FO R &M &KFNQMP  QSPQPSDJPOB MB TJHVJFOUF MJTUB JONFEJBUB EF DPOKVO UPT DFSSBEPT FO MB SFDUB SFBM B &M JOUFSWBMP DFSSBEP [a, b] QVFT TV DPNQMFNFOUP FO R FT (−∞, a) ∪ (b, +∞), FM DVBM FT BCJFSUP FO R C &M DPOKVOUP EF MPT OÙNFSPT SFBMFT Z FM DPOKVOUP WBDÎP TPO DPOKVO UPT DFSSBEPT FO R D 1BSB a ∈ R àKP MPT JOUFSWBMPT OP BDPUBEPT (−∞, a] Z [a, +∞) SFTVMUBO TFS DPOKVOUPT DFSSBEPT FO R.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

E 1BSB UPEP a ∈ R FM DPOKVOUP TJOHVMBS {a} FT VO DPOKVOUP DFSSBEP FO R ZB RVF  c {a} = (−∞, a) ∪ (a, +∞) . F &M JOUFSWBMP S = (0, 1] OP FT VO DPOKVOUP DFSSBEP FO R ZB RVF TV DPNQMFNFOUP (−∞, 0] ∪ (1, +∞) OP FT VO DPOKVOUP BCJFSUP FO R 0CTFSWBDJÓO &T JNQPSUBOUF FOGBUJ[BS RVF MPT DPODFQUPT EF DPOKVOUP BCJFSUP Z DPOKVOUP DFSSBEP OP TPO DPODFQUPT DPOUSBSJPT FT EFDJS RVF TJ VO DPOKVOUP EBEP OP FT BCJFSUP FTUP JNQMJDB RVF UFOHB RVF TFS DFSSBEP Z WJDFWFSTB DPNP TF JMVTUSÓ DPO FM DPOKVOUP (0, 1]. &KFNQMP  4FB A ⊂ R VO DPOKVOUP BCJFSUP OP WBDÎP Z BDPUBEP 1SPCFNPT RVF TVQ A ∈ / A. &GFDUJWBNFOUF TJ TVQ A ∈ A DPNP A FT VO DPOKVOUP BCJFSUP FYJTUF δ > 0 UBM RVF (TVQ A − δ, TVQ A + δ) ⊂ A. &O QBSUJDVMBS TF UJFOF RVF TVQ A + δ/2 ∈ A, MP DVBM JNQMJDB RVF TVQ A + δ/2 ≤ TVQ A, RVF FWJEFOUFNFOUF FT VOB DPOUSBEJDDJÓO &KFSDJDJP  4FBO a Z b EPT OÙNFSPT SFBMFT EJTUJOUPT 1SVFCF RVF FYJTUFO EPT BCJFSUPT EJTKVOUPT U Z V UBMFT RVF a ∈ U Z b ∈ V. &TUB QSPQJFEBE EJDF RVF FO R EPT QVOUPT TF QVFEFO TFQBSBS QPS EPT BCJFSUPT EJTKVOUPT Z TF TVFMF MMBNBS QSPQJFEBE )BVTEPSGG &KFSDJDJP  4FB F VO TVCDPOKVOUP DFSSBEP OP WBDÎP Z BDPUBEP TV QFSJPSNFOUF %FNVFTUSF RVF TVQ F ∈ F. &KFSDJDJP  4FBO G VO TVCDPOKVOUP BCJFSUP FO R Z a ∈ G 4FB {an } VOB TVDFTJÓO RVF DPOWFSHF BM SFBM a %FNVFTUSF RVF FYJTUF VO OBUVSBM N UBM RVF an ∈ G QBSB UPEP n ≥ N. "MHVOBT QSPQJFEBEFT JNQPSUBOUFT EF MPT DPOKVOUPT BCJFSUPT Z MPT DPOKVOUPT DFSSBEPT TPO QSFTFOUBEPT B DPOUJOVBDJÓO 5FPSFNB  4F DVNQMFO MBT TJHVJFOUFT QSPQJFEBEFT J 5PEB VOJÓO BSCJUSBSJB EF DPOKVOUPT BCJFSUPT FO R FT VO DPOKVOUP BCJFSUP FO R JJ 6OB JOUFSTFDDJÓO EF VO OÙNFSP àOJUP EF DPOKVOUPT BCJFSUPT FO R FT VO DPOKVOUP BCJFSUP FO R



-PT OÙNFSPT SFBMFT

JJJ 5PEB JOUFSTFDDJÓO BSCJUSBSJB EF DPOKVOUPT DFSSBEPT FO R FT VO DPO KVOUP DFSSBEP FO R JW 6OB VOJÓO EF VO OÙNFSP àOJUP EF DPOKVOUPT DFSSBEPT FO R FT VO DPOKVOUP DFSSBEP FO R ( Gi VOB VOJÓO %FNPTUSBDJÓO J 4FBO I VO DPOKVOUP EF ÎOEJDFT Z O = i∈I

BSCJUSBSJB EF DPOKVOUPT BCJFSUPT FO R 7FBNPT RVF O FT VO DPOKVOUP BCJFSUP FO R 4FB x ∈ O. 1PS EFàOJDJÓO EF VOJÓO FYJTUF j ∈ I UBM RVF x ∈ Gj Z DPNP FTUF FT VO DPOKVOUP BCJFSUP FOUPODFT FYJTUF VO OÙNFSP QPTJUJWP δx UBM RVF #δ (x) ⊂ Gj  $PNP Gj ⊆ O FOUPODFT #δ (x) ⊂ O Z BTÎ DPODMVJNPT MB QSVFCB EF J  JJ 4FBO G1 , G2 , . . . , Gk DPOKVOUPT BCJFSUPT FO MB SFDUB Z WFBNPT k  Gi FT VO DPOKVOUP BCJFSUP FO R 4J BMHVOP EF MPT Gi FT RVF G := i=1

WBDÎP MB DPODMVTJÓO FT JONFEJBUB 4VQPOHBNPT FOUPODFT RVF UPEPT MPT Gi TPO OP WBDÎPT Z TFB x ∈ G -VFHP QPS EFàOJDJÓO EF JOUFSTFDDJÓO x ∈ Gi QBSB UPEP i = 1, 2, . . . , k $PNP FTUPT TPO DPOKVOUPT BCJFSUPT FO R FYJTUFO k OÙNFSPT QPTJUJWPT δ1 , δ2 , . . . , δk DBEB VOP EFQFOEF EF x

UBM RVF #δi (x) ⊂ Gi . -B JEFB B DPOUJOVBDJÓO FT IBMMBS VO SBEJP δ UBM RVF #δ (x) ⊂ G Z QBSB FTF àO EFàOBNPT δ := NÎO {δi : i = 1, 2, . . . , k}. &T DMBSP RVF δ > 0. "àSNBNPT RVF #δ (x) ⊂ G FGFDUJWBNFOUF TJ y ∈ #δ (x) FOUPODFT QPS MB EFàOJDJÓO EFM OÙNFSP δ UFOFNPT RVF |y − x| < δ ≤ δi QBSB UPEP i = 1, 2, . . . , k MP RVF JNQMJDB RVF y ∈ #δi (x) ⊂ Gi QBSB UPEP i = 1, 2, . . . , k Z QPS UBOUP y ∈ G. 2VFEB EFNPTUSBEP JJ  JJJ &TUP FT DPOTFDVFODJB EF J Z VOB EF MBT MFZFT EF %h.PSHBO EF MB UFPSÎB EF DPOKVOUPT FM DPNQMFNFOUP EF VOB JOUFSTFDDJÓO FT MB VOJÓO EF MPT DPNQMFNFOUPT JW &TUP FT DPOTFDVFODJB EF JJ Z MB PUSB MFZ EF %h.PSHBO EF MB UFPSÎB  EF DPOKVOUPT  0CTFSWBDJÓO -B DPOEJDJÓO EF àOJUVE FO MPT ÎUFNT JJ Z JW FT DSVDJBM 1PS FKFNQMP TJ Gn = (0, 2 + 1/n) EPOEF n FT OBUVSBM Z n ≥ 1 DBEB Gn ∞  FT VO DPOKVOUP BCJFSUP FO R QFSP Gn = (0, 2] RVF DMBSBNFOUF OP FT n=1

VO DPOKVOUP BCJFSUP FO MB SFDUB 1BSB JMVTUSBS MB JNQPSUBODJB EFM OÙNFSP



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

àOJUP EF DPOKVOUPT FO FM DBTP JW

DPOTJEFSFNPT Fn = [0, 1 − 1/n] EPOEF n FT OBUVSBM Z n ≥ 1 $BEB DPOKVOUP Fn FT DFSSBEP FO R QFSP ∞ ( Fn = [0, 1) RVF DMBSBNFOUF OP FT VO DPOKVOUP DFSSBEP FO MB SFDUB n=1

&M IFDIP [0, 1) ⊆

∞ (

Fn FT DPOTFDVFODJB EF MB QSPQJFEBE BSRVJNFEJBOB

n=1

QVFT TJ 0 ≤ y < 1 FOUPODFT 1 − y > 0 Z BTÎ FYJTUF BMHÙO OBUVSBM N UBM RVF 1/N < 1 − y FTUP FT 0 ≤ y < 1 − 1/N MP DVBM JNQMJDB RVF y ∈ FN . -B PUSB JODMVTJÓO FT TJNQMF &KFNQMP  5PEP TVCDPOKVOUP àOJUP EF OÙNFSPT SFBMFT FT VO DPO KVOUP DFSSBEP FO MB SFDUB &TUP TF FYQMJDB TJNQMFNFOUF FTDSJCJFOEP EJDIP DPOKVOUP DPNP MB VOJÓO àOJUB EF MPT TJOHVMBSFT GPSNBEPT QPS TVT FMF NFOUPT Z BQMJDBOEP JW EFM UFPSFNB BOUFSJPS 3FDPSEBS UBNCJÊO RVF UPEP DPOKVOUP TJOHVMBS FT VO DPOKVOUP DFSSBEP &KFSDJDJP  4FBO F ⊆ R Z x0 ∈ F  %FNVFTUSF RVF x0 FT VO QVOUP BJTMBEP EF F TJ Z TPMP TJ F  {x0 } FT DFSSBEP %FàOJDJÓO  %FDJNPT RVF a ∈ R FT VO QVOUP JOUFSJPS EF VO TVC DPOKVOUP G ⊆ R TJ FYJTUF VOB εCPMB EF a DPOUFOJEB FO G. %FOPUBNPT QPS G◦ FM DPOKVOUP EF QVOUPT JOUFSJPSFT EFM DPOKVOUP G Z MP MMBNBNPT FM JOUFSJPS EF G. &T JONFEJBUP EF MB EFàOJDJÓO RVF G◦ FT VO DPOKVOUP BCJFSUP Z BEF NÃT G◦ ⊆ G &KFSDJDJP  )BMMF FM JOUFSJPS EF MPT TJHVJFOUFT DPOKVOUPT Q, I, [0, 1). &KFSDJDJP  %FNVFTUSF MBT TJHVJFOUFT BàSNBDJPOFT B 4FBO A Z B TVCDPOKVOUPT EF R UBMFT RVF A ⊆ B &OUPODFT A◦ ⊆ B◦. C (A ∩ B)◦ = A◦ ∩ B ◦ . y2VÊ QVFEF BàSNBS TJ TF DBNCJB ∩ QPS ∪ ) D G◦ = {A ⊂ R : A FT BCJFSUP Z A ⊆ G} &TUP TF USBEVDF EJDJFOEP RVF FM JOUFSJPS EF G FT FM BCJFSUP i NÃT HSBOEF u DPOUFOJEP FO G. E 4FB A ⊆ R VO BCJFSUP OP WBDÎP 1SVFCF RVF UPEP QVOUP EF A FT VO QVOUP EF BDVNVMBDJÓO EF A F 4FB A OP WBDÎP UBM RVF A Z A ∪ {a} TPO BCJFSUPT 1SVFCF RVF a FT VO QVOUP EF BDVNVMBDJÓO EF A



-PT OÙNFSPT SFBMFT

$PNP FT MP VTVBM FO NBUFNÃUJDBT MBT EFàOJDJPOFT TPO MBT IFSSBNJFO UBT QSJNBSJBT QBSB DBSBDUFSJ[BS MPT DPODFQUPT 4JO FNCBSHP FO NVDIPT DBTPT FTUBT TPO DPNQMJDBEBT EF WFSJàDBS Z QPS FMMP FT OFDFTBSJP QSFTFO UBS PUSBT DBSBDUFSJ[BDJPOFT RVF TFBO ÃHJMFT -BT TJHVJFOUFT TPO EF NVDIB VUJMJEBE 5FPSFNB  $BSBDUFSJ[BDJÓO EF DPOKVOUPT BCJFSUPT Z DPOKVOUPT DF SSBEPT  B 6O TVCDPOKVOUP G ⊆ R FT VO DPOKVOUP BCJFSUP FO R TJ Z TPMP TJ UPEPT TVT QVOUPT TPO JOUFSJPSFT FT EFDJS G = G◦ . C 6O TVCDPOKVOUP F ⊆ R FT VO DPOKVOUP DFSSBEP FO R TJ Z TPMP TJ DPOUJFOF UPEPT TVT QVOUPT EF BDVNVMBDJÓO FT EFDJS F  ⊆ F. D 6O TVCDPOKVOUP F ⊆ R FT VO DPOKVOUP DFSSBEP FO R TJ Z TPMP TJ UPEB TVDFTJÓO EF QVOUPT EF F RVF DPOWFSHF UJFOF MÎNJUF FO F  %FNPTUSBDJÓO B 4VQPOHBNPT RVF G FT VO DPOKVOUP BCJFSUP FO R Z WFBNPT RVF G = G◦ . 1BSB FMMP CBTUB EFNPTUSBS RVF G ⊆ G◦ . 4FB QVFT x ∈ G $PNP FTUF FT BCJFSUP FYJTUF VOB δCPMB EF x MB DVBM FTUÃ DPO UFOJEB FO G MP RVF TJHOJàDB RVF x ∈ G◦  &TUP EFNVFTUSB MB DPOEJDJÓO OFDFTBSJB MB PUSB JNQMJDBDJÓO FT TFODJMMB EF QSPCBS C 4VQPOHBNPT RVF F FT VO DPOKVOUP DFSSBEP FO MB SFDUB SFBM Z WFBNPT RVF F  ⊆ F. 3B[POFNPT QPS DPOUSBEJDDJÓO FT EFDJS TFB x VO QVOUP EF BDVNVMBDJÓO EF F FM DVBM OP FTUÃ FO F  -VFHP x ∈ F c Z DPNP FTUF FT VO DPOKVOUP BCJFSUP FYJTUF δ > 0 UBM RVF (x − δ, x + δ) ⊂ F c EF MP DVBM TF JOàFSF RVF (x − δ, x + δ) ∩ F = ∅. "IPSB CJFO DPNP x FT VO QVOUP EF BDVNVMBDJÓO EF F FOUPODFT TF EFCF DVNQMJS MB EFàOJDJÓO DPO FTUB δCPMB Z QPS UBOUP TF UJFOF RVF (x − δ, x + δ) ∩ F ∩ {x}c = ∅, MP RVF FT VOB DPOUSBEJDDJÓO $PO FTUP RVFEB EFNPTUSBEB MB JODMVTJÓO F  ⊆ F. 7FBNPT BIPSB MB PUSB JNQMJDBDJÓO TVQPOHBNPT RVF F  ⊆ F Z EFNPTUSFNPT RVF F FT VO DPOKVOUP DFSSBEP 1BSB IBDFSMP QSPCFNPT RVF TV DPNQMFNFOUP FT VO DPOKVOUP BCJFSUP 5PNFNPT x ∈ F c  1PS MB IJQÓUFTJT x OP QVFEF TFS VO QVOUP EF BDVNVMBDJÓO EF F  -VFHP FYJTUF ε > 0 UBM RVF (x − ε, x + ε) ∩ F ∩ {x}c = ∅. %F FTUP TF EFEVDF RVF (x − ε, x + ε) ⊆ {x} ∪ F c = F c MP RVF EFNVFTUSB RVF F c FT VO DPOKVOUP BCJFSUP Z QPS FOEF F FT VO DPOKVOUP DFSSBEP D 4VQPOHBNPT RVF F FT VO DPOKVOUP DFSSBEP Z TFB {xn } VOB TV DFTJÓO EF QVOUPT FO F RVF DPOWFSHF B x 1SPCFNPT RVF x ∈ F  4J OP GVFSB BTÎ FOUPODFT x FTUBSÎB FO F c Z EBEP RVF FTUF DPOKVOUP FT BCJFSUP FO MB SFDUB SFBM FYJTUJSÎB VOB CPMB EF SBEJP ε Z EF DFOUSP FO x, #ε (x) UBM RVF



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

(x − ε, x + ε) ⊂ F c .



1PS PUSB QBSUF EF MB DPOWFSHFODJB EF MB TVDFTJÓO BM QVOUP x Z QBSB FTUF ε > 0 FODPOUSBSÎBNPT VO xj ∈ F UBM RVF |xj − x| < ε 1PS  TF DPODMVJSÎB RVF xj ∈ F c MP DVBM TFSÎB VOB DPOUSBEJDDJÓO %F FTUB NBOFSB RVFEB EFNPTUSBEB MB OFDFTJEBE 1BSB EFNPTUSBS MB TVàDJFODJB VTBNPT MB DBSBDUFSJ[BDJÓO PCUFOJEB FO FM MJUFSBM BOUFSJPS 4FB y VO QVOUP EF BDVNV MBDJÓO EF F Z WFBNPT RVF y ∈ F  -VFHP QBSB UPEP OBUVSBM n DPO n ≥ 1, WÎB MB EFàOJDJÓO EF QVOUP EF BDVNVMBDJÓO DPO MBT 1/nWFDJOEBEFT

PCUFOFNPT RVF (y − 1/n, y + 1/n) ∩ F ∩ {y}c = ∅, Z QPS UBOUP QBSB UPEP n ≥ 1 FYJTUF yn ∈ F DPO yn = y Z UBM RVF |yn − y| < 1/n 0CUFOFNPT BTÎ VOB TVDFTJÓO {yn } FO F RVF DPOWFSHF BM QVOUP y 1PS OVFTUSB IJQÓUFTJT DPODMVJNPT RVF y ∈ F MP DVBM TJHOJàDB  RVF F FT VO DPOKVOUP DFSSBEP FO R.  &KFNQMP  4FB a VO QVOUP EF BDVNVMBDJÓO EF MB TVDFTJÓO {xn } 1SPCFNPT RVF FYJTUF VOB TVCTVDFTJÓO {xnk } RVF DPOWFSHF BM OÙNFSP a. #BTUB BQMJDBS MB DBSBDUFSJ[BDJÓO EF QVOUP EF BDVNVMBDJÓO EF NBOFSB SFDVSTJWB EF MB TJHVJFOUF NBOFSB FYJTUF VO FMFNFOUP EF MB TVDFTJÓO EJHBNPT xn1 UBM RVF |xn1 − a| < 1 DPO xn1 = a 1PS FM 5FPSFNB  FYJTUF VO OBUVSBM n2 > n1 UBM RVF |xn2 −a| < 1/2 Z xn2 = a 4VQPOHBNPT FTDPHJEPT n1 < n2 < · · · < nk UBMFT RVF |xnk − a| < 1/k. %BEP RVF FO 1 WFDJOEBE EF a FYJTUFO JOàOJUPT QVOUPT EF MB TVDFTJÓO QPEFNPT MB k+1 1 FTDPHFS nk+1 > nk UBM RVF |xnk+1 − a| < k+1 . %F FTUB NBOFSB IFNPT DPOTUSVJEP VOB TVCTVDFTJÓO {xnk } UBM RVF xnk → a DVBOEP k → ∞ &KFSDJDJP  4VQPOHB RVF {an } FT VOB TVDFTJÓO RVF OP UJFOF DPMB DPOTUBOUF Z FT UBM RVF an → a DVBOEP n → ∞ 1SVFCF RVF a FT FM ÙOJDP QVOUP EF BDVNVMBDJÓO EF MB TVDFTJÓO &KFSDJDJP  4FB A ⊆ R OP WBDÎP %FNVFTUSF RVF A FT VO DPOKVOUP BCJFSUP TJ Z TPMP TJ TF DVNQMF MB QSPQJFEBE TJHVJFOUF 4J {an } FT VOB TVDFTJÓO RVF DPOWFSHF B VO QVOUP EF A FOUPODFT FYJTUF VOB mDPMB EF MB TVDFTJÓO DPOUFOJEB FO A &KFSDJDJP  $POTJEFSF MBT GVODJPOFT f, g, h : R −→ R EBEBT QPS f (x) = ax + b DPO a = 0, g(x) = x2 Z h(x) = x3 . 4FB A VO BCJFSUP OP WBDÎP





%BEB VOB TVDFTJÓO {xn } MMBNBNPT mDPMB EF MB TVDFTJÓO BM DPOKVOUP {xk : k ≥ m}

-PT OÙNFSPT SFBMFT

B 1SVFCF RVF MPT DPOKVOUPT f −1 (A), g −1 (A) Z h−1 (A) TPO BCJFSUPT C 1SVFCF RVF f (A) FT VO DPOKVOUP BCJFSUP D 1SPQPSDJPOF VO FKFNQMP EF VO DPOKVOUP BCJFSUP O UBM RVF g(O) OP FT VO DPOKVOUP BCJFSUP &KFSDJDJP  5SBTMBEP EF VO BCJFSUP FT VO BCJFSUP TFBO G ⊆ R VO BCJFSUP OP WBDÎP Z c ∈ R 1SVFCF RVF FM DPOKVOUP c + G := {c + g : g ∈ G} FT BCJFSUP FO R. "EFNÃT TJ c = 0 QSVFCF RVF c · G := {c · g : g ∈ G} FT BCJFSUP FO R. &KFSDJDJP  4VNB EF DPOKVOUPT BCJFSUPT FT VO DPOKVOUP BCJFSUP TFBO A Z B DPOKVOUPT BCJFSUPT FO MB SFDUB SFBM %FNVFTUSF RVF FM DPOKVOUP A + B FT VO DPOKVOUP BCJFSUP FO R %FàOJDJÓO  4FB A ⊆ R %FàOJNPT MB DMBVTVSB P DFSSBEVSB EFM DPOKVOUP A MB DVBM EFOPUBNPT QPS A DPNP MB JOUFSTFDDJÓO EF UPEPT MPT DPOKVOUPT DFSSBEPT FO R RVF DPOUJFOFO BM DPOKVOUP A &O TÎNCPMPT  A = {F ⊆ R : F FT DFSSBEP Z F ⊇ A}. 0USB GPSNB EF EFDJS FTUP FT RVF MB DMBVTVSB EF A FT FM iNFOPSu DFSSBEP RVF DPOUJFOF BM DPOKVOUP A %FTUBDBNPT RVF EF MB QBSUF JJJ EFM 5FPSFNB  TF TJHVF RVF A FT VO DPOKVOUP DFSSBEP FO R &O FM QSÓYJNP UFPSFNB TF SFÙOFO BMHVOBT QSPQJFEBEFT Z DBSBDUFSJ [BDJPOFT EF MB DMBVTVSB EF VO DPOKVOUP 5FPSFNB  4FB A ⊆ R &OUPODFT J -B DMBVTVSB EF A DPOUJFOF BM DPOKVOUP A FT EFDJS A ⊆ A. JJ &M DPOKVOUP A FT DFSSBEP TJ Z TPMP TJ A = A. &O QBSUJDVMBS A = A. JJJ 4J A ⊆ B FOUPODFT A ⊆ B. JW 4F WFSJàDB RVF A ∪ B = A ∪ B. W x ∈ A TJ Z TPMP TJ UPEB εCPMB EF x DPOUJFOF QVOUPT EFM DPOKVOUP A TJ Z TPMP TJ FYJTUF VOB TVDFTJÓO EF QVOUPT FO A RVF DPOWFSHF B x WJ x ∈ A TJ Z TPMP TJ ÎOG {|x − a| : a ∈ A} = 0.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

%FNPTUSBDJÓO J 4JNQMFNFOUF VTBNPT MB EFàOJDJÓO QVFT TJ x ∈ A FOUPODFT x QFSUFOFDF B DVBMRVJFS DFSSBEP F RVF DPOUFOHB BM DPOKVOUP A MP DVBM RVJFSF EFDJS RVF x ∈ A. JJ 4VQPOHBNPT RVF A FT DFSSBEP Z QSPCFNPT RVF A = A. 1PS J

CBTUB WFS MB JODMVTJÓO A ⊆ A TFB QVFT x ∈ A. -VFHP x QFSUFOFDF B UPEPT MPT DPOKVOUPT DFSSBEPT RVF DPOUJFOFO B A Z DPNP A FT DFSSBEP Z A ⊇ A FOUPODFT x ∈ A &TUP QSVFCB MB DPOEJDJÓO OFDFTBSJB -B TVàDJFODJB FT JONFEJBUB JJJ $PNP B ⊇ B ⊇ A, TJ x ∈ A FOUPODFT x QFSUFOFDF B UPEPT MPT DPOKVOUPT DFSSBEPT RVF DPOUJFOFO B A TJFOEP B VOP EF FMMPT -VFHP A ⊆ B. JW :B RVF UBOUP A DPNP B FTUÃO JODMVJEPT FO A ∪ B MB JODMVTJÓO A∪B ⊆A∪B FT DPOTFDVFODJB EFM ÎUFN QSFWJP 1BSB MB JODMVTJÓO SFDÎQSPDB UPNBNPT x ∈ A ∪ B Z QPS UBOUP x ∈ F QBSB UPEP DPOKVOUP F DFSSBEP RVF DPOUFOHB MB VOJÓO EF A Z B FO QBSUJDVMBS x QFSUFOFDF BM DFSSBEP F := A ∪ B, FM DVBM DPOUJFOF B A ∪ B. "TÎ RVFEB EFNPTUSBEP FTUF ÎUFN W 7BNPT B EFNPTUSBS FM DJDMP QSJNFSB BàSNBDJÓO ⇒ TFHVOEB BàS NBDJÓO ⇒ UFSDFSB BàSNBDJÓO ⇒ QSJNFSB BàSNBDJÓO 4FBO x ∈ A Z TV QPOHBNPT QPS FM BCTVSEP RVF FYJTUF VOB δCPMB EF x EJHBNPT #δ (x) ≡ # RVF OP DPOUJFOF QVOUPT EF A FTUP FT A ∩ #δ (x) = ∅. &TUP JNQMJDBSÎB RVF A ⊆ #c , FM DVBM FT VO DPOKVOUP DFSSBEP -VFHP QPS EFàOJDJÓO EF DMBVTVSB UFOESÎBNPT RVF x ∈ / # RVF DMBSBNFOUF FT VOB DPOUSBEJDDJÓO "IPSB WFBNPT FM TFHVOEP DJDMP BQMJDBOEP MB IJQÓUFTJT DPO ε = 1/n FODPOUSBNPT RVF QBSB DBEB OBUVSBM n ≥ 1, #1/n (x) DPO UJFOF QVOUPT EF A MP RVF TJHOJàDB RVF FYJTUF xn ∈ A UBM RVF xn → x. 'JOBMNFOUF QSPCFNPT RVF MB UFSDFSB BàSNBDJÓO JNQMJDB MB QSJNFSB TFB F VO DPOKVOUP DFSSBEP FO R RVF DPOUJFOF BM DPOKVOUP A Z QSPCFNPT RVF x ∈ F  $PNP FYJTUF VOB TVDFTJÓO EF QVOUPT FO A RVF DPOWFSHF B x FTUB SFTVMUB DPOUFOJEB FO F Z QPS D EFM 5FPSFNB  TF DPODMVZF RVF x ∈ F  %BEP RVF FM DPOKVOUP DFSSBEP F FT BSCJUSBSJP PCUFOFNPT RVF x ∈ A. WJ 4VQPOHBNPT RVF x ∈ A. 1PS MP EFNPTUSBEP QSFWJBNFOUF FYJTUF VOB TVDFTJÓO EF QVOUPT xn ∈ A UBM RVF xn → x. %JDIP EF PUSB GPSNB UFOFNPT RVF |xn − x| → 0 DVBOEP n → ∞ "IPSB CJFO TJ EFOPUBNPT m := ÎOG {|x − a| : a ∈ A} FT DMBSP RVF 0 ≤ m ≤ |xn − x| QBSB UP EP OBUVSBM n )BDJFOEP RVF n → ∞ FO FTUBT EFTJHVBMEBEFT PCUFOFNPT



-PT OÙNFSPT SFBMFT

RVF m = 0. 3FDÎQSPDBNFOUF TVQPOHBNPT RVF m = 0 Z WFBNPT RVF x ∈ A. 1BSB FMMP QSPCFNPT RVF FO UPEB εWFDJOEBE EF x FYJTUFO FMFNFO UPT EF A 4FB ε > 0 1PS DBSBDUFSJ[BDJÓO EF ÎOàNP FYJTUF a ∈ A UBM RVF ε > |x − a | FTUP FT a ∈ #ε (x) Z BTÎ UFOFNPT RVF A ∩ #ε (x) = ∅. 1PS UBOUP DVBMRVJFS εCPMB EF x DPOUJFOF QVOUPT EF A MP DVBM RVJFSF EFDJS   RVF x ∈ A. &TUP DPNQMFUB MB QSVFCB EFM UFPSFNB /PUFNPT RVF EF MB QSJNFSB FRVJWBMFODJB FO W TF TJHVF RVF A ⊆ A. &KFSDJDJP  4FB A ⊆ R %FNVFTUSF RVF A = A ∪ A . &KFSDJDJP  %FNVFTUSF x ∈ A◦ ⇐⇒ ÎOG {|x − y| : y ∈ Ac } > 0. &KFSDJDJP  1SVFCF RVF G FT BCJFSUP TJ Z TPMP TJ (∀ A ⊆ R) (G ∩ A ⊆ G ∩ A). 5FPSFNB  4FB A ⊆ R &OUPODFT FM DPOKVOUP EF QVOUPT EF BDVNV MBDJÓO EF A, A FT VO DPOKVOUP DFSSBEP %FNPTUSBDJÓO #BTUB QSPCBS RVF (A ) ⊆ A . 4FB x ∈ (A ) Z TVQPOHB NPT MP DPOUSBSJP FT EFDJS RVF x ∈ / A . &OUPODFT FYJTUJSÎB δ > 0 UBM RVF #δ (x) ∩ A ⊆ {x}. 1PS PUSB QBSUF UFOFNPT RVF #δ (x) ∩ A FT VO DPOKVOUP JOàOJUP Z DPNP A ⊆ A FOUPODFT #δ (x) ∩ A UBNCJÊO FT VO DPOKVOUP JOàOJUP 1PS FM &KFSDJDJP  BQMJDBEP DPO G = #δ (x) PCUFOESÎBNPT RVF #δ (x) ∩ A ⊆ #δ (x) ∩ A ⊆ {x} = {x}, EF MP DVBM TF DPODMVJSÎB RVF #δ (x) ∩ A TFSÎB VO DPOKVOUP àOJUP &TUB DPOUSBEJDDJÓO EFNVFTUSB FM  UFPSFNB  %FàOJDJÓO  4FB A ⊆ R %FDJNPT RVF a ∈ R FT VO QVOUP GSPOUFSB EFM DPOKVOUP A TJ UPEB εCPMB EF a DPOUJFOF QVOUPT EF A Z EF Ac  &M DPOKVOUP EF QVOUPT GSPOUFSB EF A MP EFOPUBNPT QPS ∂A Z MP MMB NBNPT MB GSPOUFSB EF A &T TFODJMMP WFS RVF MB EFàOJDJÓO EF GSPOUFSB EF VO DPOKVOUP FT FRVJ WBMFOUF B MB JHVBMEBE ∂A = A ∩ Ac . &O DPOTFDVFODJB MB GSPOUFSB EF VO TVCDPOKVOUP A EF R FT VO DPOKVOUP DFSSBEP FO R -BT OPDJPOFT EF DMBVTVSB Z GSPOUFSB EF VO DPOKVOUP FTUÃO SFMBDJPOB EBT DPNP TF NVFTUSB B DPOUJOVBDJÓO 5FPSFNB  4FB A ⊆ R &OUPODFT B &M DPOKVOUP A FT DFSSBEP TJ Z TPMP TJ ∂A ⊆ A.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

C A = A ∪ ∂A. %FNPTUSBDJÓO B 4VQPOHBNPT RVF A FT DFSSBEP &OUPODFT ∂A = A ∩ Ac = A ∩ Ac ⊆ A. 3FDÎQSPDBNFOUF TVQPOHBNPT RVF ∂A ⊆ A Z WFBNPT RVF A ⊆ A DPO MP DVBM TF DPODMVZF RVF A FT DFSSBEP &O FGFDUP FTUP FT JONFEJBUP QVFT TJ x ∈ A Z x OP QFSUFOFDJFSB BM DPOKVOUP A FOUPODFT x ∈ Ac ⊆ Ac Z QPS UBOUP x ∈ ∂A ⊆ A RVF QPS TVQVFTUP FT VOB DPOUSBEJDDJÓO C A ∪ ∂A = A ∪ (A ∩ Ac ) = (A ∪ A) ∩ (A ∪ Ac ) = A ∩ (A ∪ Ac ) = A ∩ R = A. -B QFOÙMUJNB JHVBMEBE TF UJFOF QPSRVF R ⊇ A ∪ Ac ⊇ A ∪ Ac = R.   &KFSDJDJP  1SVFCF RVF x ∈ ∂A TJ Z TPMP TJ FYJTUFO TVDFTJPOFT {xn } FO A Z {yn } FO Ac SFTQFDUJWBNFOUF UBMFT RVF xn −→ x DVBOEP n → ∞ Z yn −→ x DVBOEP n → ∞ &KFSDJDJP  &ODVFOUSF MB DMBVTVSB FO R EF MPT TJHVJFOUFT DPOKVOUPT I, Z, Q Z MB TVDFTJÓO {1/n} &KFSDJDJP  %FUFSNJOF MB WFSBDJEBE P GBMTFEBE EF MBT BàSNBDJPOFT 4J A ⊆ B FOUPODFT ∂A ⊆ ∂B. ∂A = ∂Ac . &KFSDJDJP  1SVFCF MBT BàSNBDJPOFT B -B GSPOUFSB EF VO DPOKVOUP FT WBDÎB TJ Z TPMP TJ EJDIP DPOKVOUP FT BCJFSUP Z DFSSBEP C 6O DPOKVOUP FT BCJFSUP TJ Z TPMP TJ OP DPOUJFOF QVOUPT EF TV GSPO UFSB D 1BSB UPEP TVCDPOKVOUP A EF R TF DVNQMF RVF Ac = (A◦ )c . E 1BSB UPEP TVCDPOKVOUP A EF R TF DVNQMF RVF ∂A = Ac  A◦ . F 4J F ⊆ R FT VO DPOKVOUP DFSSBEP FOUPODFT (∂F )◦ = ∅. &KFSDJDJP  1SVFCF TJ A ⊂ R FT OP WBDÎP Z BDPUBEP TVQFSJPSNFOUF FOUPODFT TVQ A ∈ ∂A. &TUBCMF[DB VO SFTVMUBEP BOÃMPHP QBSB VO DPOKVOUP BDPUBEP JOGFSJPSNFOUF



-PT OÙNFSPT SFBMFT

&KFSDJDJP  %FNVFTUSF MBT BàSNBDJPOFT B ∂(S1 ∪ S2 ) ⊆ ∂S1 ∪ ∂S2 . C ∂(S1 ∩ S2 ) ⊆ ∂S1 ∪ ∂S2 . D ∂(S1  S2 ) ⊆ ∂S1 ∪ ∂S2 . E ∂(S) ⊆ ∂S.

 $POKVOUPT DPNQBDUPT FO MB SFDUB SFBM -B OPDJÓO EF DPNQBDJEBE FT VOB EF MBT NÃT ÙUJMFT FO FM BOÃMJTJT NB UFNÃUJDP 4PO WBSJBEBT TVT BQMJDBDJPOFT QPS DJUBS VO QBS EF FKFNQMPT NFODJPOFNPT RVF QFSNJUF HBSBOUJ[BS MB FYJTUFODJB EF QVOUPT EF NÃ YJNP P NÎOJNP EF VOB GVODJÓO DPOUJOVB FO VO JOUFSWBMP DFSSBEP Z MB PCUFODJÓO EF SFTVMUBEPT EF FYJTUFODJB EF TPMVDJPOFT EF DJFSUP UJQP EF FDVBDJPOFT MMBNBEBT FDVBDJPOFT EJGFSFODJBMFT %FàOJDJÓO  4FBO J = ∅ VO DPOKVOUP EF ÎOEJDFT Z Oα ⊆ R VO BCJFSUP QBSB DBEB α ∈ J. %FDJNPT RVF MB DPMFDDJÓO {Oα : α ∈ J} FT VO DVCSJNJFOUP BCJFSUP QBSB VO DPOKVOUP A ⊆ R TJ ( A⊆ Oα . α∈J

4J I ⊂ J EFDJNPT RVF {Oα : α ∈ I} FT VO TVCDVCSJNJFOUP EFM DPOKVOUP ( A TJ A ⊆ Oα . 4J I ⊂ J FT àOJUP EFDJNPT RVF FM TVCDVCSJNJFOUP FT àOJUP

α∈I

&KFNQMP  -B DPMFDDJÓO {(0, n) : n ∈ N} FT VO DVCSJNJFOUP BCJFS UP QBSB A := (0, +∞) ZB RVF EBEP x ∈ A FYJTUF N ∈ N UBM RVF 0 < x < N Z BTÎ x ∈ (0, N ). "RVÎ FM DPOKVOUP EF ÎOEJDFT FT J = N. 6O TVCDVCSJNJFOUP QBSB A FT {(0, n) : n ∈ 2N} FO FTUF DBTP FM TVCDPOKVOUP EF ÎOEJDFT FT I = 2N FM DPOKVOUP EF OBUVSBMFT QBSFT n ) : n ∈ N} FT VO DVCSJNJFOUP BCJFS 7FBNPT RVF MB DPMFDDJÓO {(0, n+1 UP QBSB A := (0, 1). 4J 0 < x < 1 FOUPODFT QPS MB QSPQJFEBE BSRVJNF EJBOB FYJTUF VO OBUVSBM N UBM RVF 1 − x > 1/N > 1/(N + 1) Z QPS ∞ ( (0, n/(n + 1)). UBOUP x < N /(N + 1). &TUP JNQMJDB RVF (0, 1) ⊆ n=0

%FàOJDJÓO  4FB K ⊂ R %FDJNPT RVF K FT VO DPOKVOUP DPNQBDUP TJ DVBMRVJFS DVCSJNJFOUP BCJFSUP EF K BENJUF VO TVCDVCSJNJFOUP àOJUP QBSB K



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP  &T JONFEJBUP EF MB EFàOJDJÓO RVF FM DPOKVOUP WBDÎP FT DPNQBDUP .ÃT BÙO DVBMRVJFS DPOKVOUP àOJUP FT VO DPOKVOUP DPNQBDUP $PNP VOB UFSDFSB JMVTUSBDJÓO WFBNPT RVF FM JOUFSWBMP (0, 1) OP FT VO DPOKVOUP DPNQBDUP 1BSB FMMP EFCFNPT FYIJCJS VO DVCSJNJFOUP BCJFSUP EF (0, 1) UBM RVF UPEP TVCDVCSJNJFOUP àOJUP OP DVCSF B EJDIP DPOKVOUP 6TBSFNPT FM DVCSJNJFOUP EFM &KFNQMP  %FOPUFNPT QPS Vn DBEB BCJFSUP EFM DVCSJNJFOUP Z TVQPOHBNPT QPS DPOUSBEJDDJÓO RVF FYJTUF BMHÙO TVCDVCSJNJFOUP àOJUP RVF DVCSF B (0, 1) FTUP FT TVQPOHBNPT RVF FYJTUFO OBUVSBMFT n1 , n2 , . . . , np UBMFT RVF (0, 1) ⊆

p (

Vi .

i=1

4FB M := NÃY{n1 , n2 , . . . , np }. $PNP {Vn } FT VOB DPMFDDJÓO DSFDJFOUF FOUPODFT TF UFOESÎB RVF (0, 1) ⊆ VM = (0, M /(M + 1)). &M BCTVSEP TF UJFOF BM PCTFSWBS RVF (M + 1)/(M + 2) ∈ (0, 1) QFSP OP FTUÃ FO VM  &TUB DPOUSBEJDDJÓO NVFTUSB RVF FM DPOKVOUP (0, 1) OP FT DPNQBDUP " QSJNFSB WJTUB MB EFàOJDJÓO EF DPNQBDJEBE RVF IFNPT QSFTFOUBEP FT RVJ[Ã QPDP JOUVJUJWB *OUSPEVDJSFNPT GPSNBT FRVJWBMFOUFT EF FYQSFTBS FTUF DPODFQUP EF NBOFSB RVF TFB DPNQMFUBNFOUF BNJHBCMF B OVFTUSB JOUVJDJÓO Z BEFNÃT EF GÃDJM JOUFSJPSJ[BDJÓO &O FTB EJSFDDJÓO FNQF[BNPT DPO FM TJHVJFOUF SFTVMUBEP RVF QFSNJUF NPTUSBS DMBSBNFOUF FM QPS RVÊ EF MB SFTQVFTUB OFHBUJWB FO FM BOUFSJPS FKFNQMP Z NÃT BÙO FT VO DBTP QBSUJDVMBS EF VO IFDIP HFOFSBM RVF QSFTFOUBSFNPT NÃT BEFMBOUF 5FPSFNB  4FBO a Z b OÙNFSPT SFBMFT DPO a < b. &OUPODFT FM JOUFSWBMP [a, b] FT VO DPOKVOUP DPNQBDUP %FNPTUSBDJÓO -B QSVFCB FT JMVTUSBUJWB FO FM TFOUJEP EF RVF TF QVFEF DPOTJEFSBS DPNP VOB UÊDOJDB MB DVBM DPOTJTUF FO DPOTUSVJS VO DPOKVOUP RVF SFÙOB MPT FMFNFOUPT DPO MBT QSPQJFEBEFT RVF VOP EFTFB 4FB {Vα } DPO α ∈ J VO DVCSJNJFOUP BCJFSUP QBSB [a, b] %FàOBNPT FM DPOKVOUP EF UPEPT MPT JOUFSWBMPT EF MB GPSNB [a, x] RVF TF QVFEFO DVCSJS DPO VO TVCDVCSJNJFOUP àOJUP EF {Vα } FT EFDJS * ( E := x ∈ [a, b] : [a, x] ⊆ Vα QBSB BMHÙO I ⊂ J àOJUP . α∈I

%FNPTUSFNPT RVF b ∈ E, MP DVBM UFSNJOB MB QSVFCB EFM UFPSFNB 1BSB FMMP VTBSFNPT MB DPNQMFUF[ EF MPT OÙNFSPT SFBMFT Z QSPQJFEBEFT EFM TVQSFNP *OJDJBMNFOUF OPUFNPT RVF E FT OP WBDÎP QVFT a ∈ E : [a, a] = {a} Z FYJTUF β ∈ J UBM RVF a ∈ Vβ . "EJDJPOBMNFOUF FM DPOKVOUP E FTUÃ BDPUBEP



-PT OÙNFSPT SFBMFT

TVQFSJPSNFOUF QPS b -VFHP FYJTUF c ∈ R UBM RVF c = TVQ E 0CTFSWFNPT UBNCJÊO RVF a ≤ c ≤ b Z QPS UBOUP FYJTUF λ ∈ J UBM RVF c ∈ Vλ . $PNP FTUF ÙMUJNP FT VO DPOKVOUP BCJFSUP FYJTUF ε > 0 UBM RVF (c−ε, c+ε) ⊂ Vλ  1PS PUSB QBSUF FYJTUF x0 ∈ E UBM RVF c − ε < x0 Z EF FTUB NBOFSB UFOFNPT RVF m ( V αi . [a, x0 ] ⊆ i=1

$PNP c + ε/2 ∈ / E FTUF OÙNFSP OP WFSJàDB QPS MP NFOPT VOB EF MBT DPOEJDJPOFT RVF DBSBDUFSJ[B MPT FMFNFOUPT EF E 1FSP [a, c + ε/2] = [a, x0 ] ∪ [x0 , c + ε/2] ⊆

m (

Vαi ∪ Vλ ,

i=1

FOUPODFT OFDFTBSJBNFOUF c + ε/2 ∈ / [a, b]. &TUP JNQMJDB RVF c + ε/2 < a P CJFO c + ε/2 > b -B QSJNFSB OP QVFEF PDVSSJS ZB RVF TF PCUFOESÎB c + ε/2 < a ≤ c MP DVBM BSSPKBSÎB MB DPOUSBEJDDJÓO ε < 0. 1PS UBOUP UJFOF RVF EBSTF MB TFHVOEB PQDJÓO FTUP FT c + ε/2 > b EF FTUP TF TJHVF RVF m ( V αi ∪ V λ . [a, b] ⊂ [a, c + ε/2] ⊆ i=1

$PO FTUP RVFEB EFNPTUSBEP RVF b ∈ E.

 

&KFSDJDJP  %FNVFTUSF VTBOEP MB EFàOJDJÓO RVF FM DPOKVOUP [0, +∞) OP FT DPNQBDUP &KFSDJDJP  4FB {an } VOB TVDFTJÓO FO R UBM RVF an → a 4J K = {an } ∪ {a}, QSVFCF RVF K FT DPNQBDUP &KFSDJDJP  $POTJEFSF MBT GVODJPOFT EFM &KFSDJDJP  Z TFB K ⊂ R VO DPNQBDUP %FNVFTUSF RVF f (K), g(K) Z h(K) TPO DPOKVOUPT DPN QBDUPT 4VHFSFODJB FM ÎUFN B EFM DJUBEP FKFSDJDJP QVFEF BZVEBS

5FPSFNB  4FBO A Z K TVCDPOKVOUPT EF R UBMFT RVF A FT DFSSBEP K FT DPNQBDUP Z A ⊆ K. &OUPODFT A FT DPNQBDUP %FNPTUSBDJÓO 4FB {Vα } DPO α ∈ J VO DVCSJNJFOUP BCJFSUP QBSB A Z WFBNPT RVF FT QPTJCMF FYUSBFS VO TVCDVCSJNJFOUP àOJUP QBSB EJDIP DPO ) c QPS MB DPNQBDJEBE KVOUP %BEP RVF K ⊆ R = A ∪ A ⊆ α∈J (Vα ∪ Ac ) ) EF K FYJTUF VO TVCDPOKVOUP àOJUP I ⊂ J UBM RVF K ⊆ α∈I (Vα ∪Ac ) %F )  FTUP TF TJHVF RVF A ⊆ α∈I Vα MP DVBM TJHOJàDB RVF A FT DPNQBDUP  

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP  &M DPOKVOUP RVF EFTDSJCJNPT B DPOUJOVBDJÓO FT EF TVNB JNQPSUBODJB FO NBUFNÃUJDBT Z UJFOF QSPQJFEBEFT JODSFÎCMFT QFSP TPMB NFOUF EFTUBDBSFNPT VO QBS EF FMMBT 4FB F0 = [0, 1] %JWJEJNPT FTUF FO USFT UFSDJPT EF JHVBM MPOHJUVE Z EFTDBSUBNPT FM UFSDJP NFEJP BCJFSUP

RVFEÃOEPOPT DPO F1 = [0, 1/3] ∪ [2/3, 1]. $BEB VOP EF MPT JOUFSWBMPT RVF DPOGPSNBO F1 MP EJWJEJNPT FO UFSDJPT JHVBMFT Z EFTDBSUBNPT MPT UFSDJPT DFOUSBMFT BCJFSUPT  "TÎ FO FTUB TFHVOEB FUBQB PCUFOFNPT F2 = [0, 1/32 ] ∪ [2/32 , 3/32 ] ∪ [6/32 , 7/32 ] ∪ [8/32 , 1]. &M QSPDFTP TF SFQJUF Z FO MB nÊTJNB FUBQB PCUFOFNPT Fn DPNP VOJÓO EF 2n JOUFSWBMPT EF MPOHJUVE 1/3n DBEB VOP %FàOJNPT FM DPOKVOUP EF $BOUPS EF MB TJHVJFOUF NBOFSB C :=



Fn .

n∈N

0CTFSWFNPT RVF C = ∅ QVFT 0, 1 ∈ C JODMVTP MPT FYUSFNPT EF MPT JOUFS WBMPT UFSDJPT EFTDBSUBEPT QFSUFOFDFO BM DPOKVOUP EF $BOUPS 'JOBMNFOUF DPNP DBEB Fn FT DFSSBEP FOUPODFT FM DPOKVOUP EF $BOUPS FT DFSSBEP Z BEFNÃT FT DPNQBDUP QVFT F0 MP FT Z C ⊆ F0 . -B PUSB QSPQJFEBE RVF NFO DJPOBSFNPT FT RVF FM DPOKVOUP EF $BOUPS UJFOF JOUFSJPS WBDÎP TJ OP GVFSB BTÎ FYJTUJSÎB x0 ∈ C ◦ FT EFDJS FYJTUJSÎB JO JOUFSWBMP BCJFSUP (a, b) ⊂ C Z QPS UBOUP (a, b) ⊂ Fn QBSB UPEP OBUVSBM n %F FTUP TF TFHVJSÎB RVF 0 < b − a ≤ (2/3)n QBSB UPEP OBUVSBM n Z BTÎ UFOESÎBNPT RVF b = a MP DVBM FT BCTVSEP %F FTUB NBOFSB DPODMVJNPT RVF FM DPOKVOUP EF $BOUPS OP UJFOF QVOUPT JOUFSJPSFT "IPSB QSFTFOUBNPT VO SFTVMUBEP NÃT HFOFSBM RVF FM 5FPSFNB  5FPSFNB  5PEP TVCDPOKVOUP DFSSBEP Z BDPUBEP EF R FT DPNQBDUP %FNPTUSBDJÓO 4FB A VO TVCDPOKVOUP EF R DFSSBEP Z BDPUBEP 1PS FM BDPUBNJFOUP FYJTUF VOB DPOTUBOUF M > 0 UBM RVF QBSB UPEP x ∈ A TF DVNQMF |x| ≤ M  &O PUSBT QBMBCSBT A ⊆ [−M, M ] %FCJEP B MB DPNQB DJEBE EF [−M, M ] 5FPSFNB  Z B MB DFSSBEVSB EF A DPODMVJNPT  RVF A FT DPNQBDUP  &M SFDÎQSPDP EF FTUF UFPSFNB UBNCJÊO FT DJFSUP 5FPSFNB  4FB K ⊂ R VO DPOKVOUP DPNQBDUP &OUPODFT K FT DFSSBEP Z BDPUBEP



-PT OÙNFSPT SFBMFT

%FNPTUSBDJÓO 1VFTUP RVF {(−n, n) : n ∈ N} FT VO DVCSJNJFOUP BCJFSUP QBSB R ⊃ K QPS MB IJQÓUFTJT FYJTUF VO DPOKVOUP àOJUP EF OBUVSBMFT {n1 , n2 , . . . , np } UBM RVF p (

K⊆

(−ni , ni ).

i=1

4J M := NÃY{n1 , n2 , . . . , np } FOUPODFT SFTVMUB RVF K ⊆ (−M, M ) Z QPS UBOUP K FT BDPUBEP 7FBNPT BIPSB RVF K FT DFSSBEP 1BSB FTUP QSPCFNPT RVF K c FT VO DPOKVOUP BCJFSUP TFB y ∈ K c  1BSB DBEB OBUVSBM n ≥ 1 FM DPOKVOUP Gn := (−∞, y−1/n)∪(y+1/n, +∞) FT VO DPOKVOUP ∞ ( Gn ZB RVF TJ w ∈ K FOUPODFT BCJFSUP Z BEFNÃT TF DVNQMF RVF K ⊆ n=1

|w − y| > 0 QPS UBOUP FYJTUF VO OBUVSBM N UBM RVF 1/N < w − y P CJFO 1/N < y − w MP RVF RVJFSF EFDJS RVF w ∈ GN . 1PS MB DPNQBDJEBE EFM DPOKVOUP K QPEFNPT SFEVDJS FM DVCSJNJFOUP B VO TVCDVCSJNJFOUP àOJUP FT EFDJS m ( Gni . K⊆ i=1

$PNP BOUFT TJ L := NÃY{ni : i = 1, 2, . . . , m} FOUPODFT K ⊆ GL RVF FT FRVJWBMFOUF B EFDJS RVF GcL ⊆ K c MP DVBM JNQMJDB RVF (y −

1 2L ,

y+

1 2L )

⊂ K c.

&TUP EFNVFTUSB RVF K c FT BCJFSUP Z QPS UBOUP K FT DFSSBEP

 

-PT EPT ÙMUJNPT UFPSFNBT TF DPOKVHBO FO VOP TPMP FM DVBM FT VO SFTVMUBEP DMÃTJDP EF DPNQBDJEBE 5FPSFNB  5FPSFNB EF )FJOF#PSFM  6O DPOKVOUP K ⊂ R FT DPN QBDUP TJ Z TPMP TJ K FT DFSSBEP Z BDPUBEP &KFSDJDJP  4FB K ⊂ R Z OP WBDÎP %FNVFTUSF K FT DPNQBDUP TJ Z TPMP TJ UPEB TVDFTJÓO FO K UJFOF BM NFOPT VOB TVCTVDFTJÓO RVF DPOWFSHF FO K.

 -ÎNJUFT Z DPOUJOVJEBE FO MB SFDUB )BTUB BIPSB MB OPDJÓO EF MÎNJUF OPT FT DPOPDJEB QBSB VOB DMBTF QBSUJDVMBS EF GVODJÓO DVZP EPNJOJP FT VO TVCDPOKVOUP EF MPT OÙNFSPT OBUVSBMFT VOB TVDFTJÓO &M DPODFQUP EF MÎNJUF UBNCJÊO TF QSFTFOUB QBSB GVODJPOFT NÃT HFOFSBMFT DPO EPNJOJP VO TVCDPOKVOUP EF R



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

%FàOJDJÓO  4FBO f : A ⊆ R −→ R VOB GVODJÓO Z c ∈ R VO QVOUP EF BDVNVMBDJÓO EF A %FDJNPT RVF L ∈ R FT VO MÎNJUF EF f FO c, TJ QBSB UPEP ε > 0 FYJTUF VO δ > 0 RVF WFSJàDB MB TJHVJFOUF DPOEJDJÓO (∀ x) (x ∈ A ∧ 0 < |x − c| < δ ⇒ |f (x) − L| < ε). 6TBSFNPT MBT OPUBDJPOFT MÎN f (x) = L P UBNCJÊO f (x) −→ L DVBO x→c EP x → c. 0CTFSWFNPT RVF FTUB EFàOJDJÓO TF GPSNVMB EF NBOFSB FRVJWBMFOUF BTÎ MÎN f (x) = L TJ Z TPMP TJ QBSB UPEB εCPMB #ε (L) EF L FYJTUF VOB x→c

δCPMB #δ (c) EF c UBM RVF (∀ x) (x ∈ A ∩ #δ (c) ∧ x = c ⇒ f (x) ∈ #ε (L)). 5FPSFNB  4J FM MÎNJUF EF VOB GVODJÓO f : A ⊆ R −→ R FYJTUF FO VO QVOUP EF BDVNVMBDJÓO c ∈ R EF A FOUPODFT FT ÙOJDP %FNPTUSBDJÓO 4VQPOHBNPT RVF FYJTUFO SFBMFT L1 = L2 UBMFT RVF MÎN f (x) = L1 Z MÎN f (x) = L2 .

x→c

x→c

&OUPODFT QBSB ε = 12 |L1 − L2 | > 0 FYJTUFO δ1 > 0 Z δ2 > 0 UBMFT RVF (∀ x) (x ∈ A ∧ 0 < |x − c| < δ1 ⇒ |f (x) − L1 | < ε)



(∀ x) (x ∈ A ∧ 0 < |x − c| < δ2 ⇒ |f (x) − L2 | < ε).



Z

4FB δ = NÎO {δ1 , δ2 } > 0 1PS UBOUP QBSB UPEP x ∈ A DPO 0 < |x−c| < δ MBT JNQMJDBDJPOFT EF  Z  TF DVNQMFO Z BTÎ TF UJFOF RVF |f (x) − L1 | < ε Z |f (x) − L2 | < ε. -VFHP QBSB MPT x ∈ A DPO 0 < |x − c| < δ TF JOàFSF RVF |L1 − L2 | ≤ |L1 − f (x)| + |f (x) − L2 | < 2ε = |L1 − L2 |, RVF FWJEFOUFNFOUF FT VOB DPOUSBEJDDJÓO -VFHP FM MÎNJUF FT ÙOJDP

 

&KFNQMP  4J f FT VOB GVODJÓO DPOTUBOUF EJHBNPT f (x) = b QBSB UPEP x ∈ A Z c ∈ A FOUPODFT MÎN b = b. x→c



-PT OÙNFSPT SFBMFT

&KFNQMP  4FBO f : (0, +∞) −→ R EBEB QPS f (x) = x1 Z c > 0 àKP $PNP FM JOUFSWBMP (0, +∞) FT VO DPOKVOUP BCJFSUP Z c ∈ (0, +∞) FOUPODFT c FT VO QVOUP EF BDVNVMBDJÓO EF EJDIP JOUFSWBMP 7FBNPT RVF MÎN

1

x→c x

1 = · c

4FB ε > 0 Z FODPOUSFNPT δ > 0 UBM RVF TJ x > 0 Z 0 < |x − c| < δ FOUPODFT |1/x − 1/c| < ε. /PUFNPT RVF |

|x − c| 1 1 − |= , x c cx



TFSÃ NFOPS RVF ε TJ MPHSBNPT BDPUBS JOGFSJPSNFOUF FM EFOPNJOBEPS RVF JOUVJUJWBNFOUF FTUÃ DFSDBOP B c2 QBSB UPEPT MPT x > 0 DFSDBOPT B c 4VQPOHBNPT RVF 0 < |x − c| < δ DPO δ > 0 B EFUFSNJOBS Z USBUFNPT FOUPODFT EF BDPUBS JOGFSJPSNFOUF FM EFOPNJOBEPS x = c − (c − x) ≥ c − |x − c| > c − δ. &TUP OPT TVHJFSF SFTUSJOHJS δ > 0 EF NBOFSB RVF δ < c QPS FKFNQMP δ ≤ c/2 "TÎ MBT DPTBT x > c/2 QBSB UPEP x > 0 DPO |x − c| < δ Z δ ≤ c/2 -MFWBOEP FTUP BM FTUJNBUJWP  MMFHBNPT B RVF |

2δ 1 1 − | < 2, x c c

MP DVBM TFSÃ NFOPS RVF ε > 0 TJ 2δ/c2 ≤ ε &O SFTVNFO EBEP ε > 0 UPNBNPT δ > 0 DPNP δ = NÎO {c/2, εc2 /2} EF NBOFSB RVF TJ x > 0 DPO 0 < |x − c| < δ FOUPODFT |1/x − 1/c| < ε &TUP EFNVFTUSB RVF 1/x −→ 1/c DVBOEP x → c. x−4 · &M &KFNQMP  1SPCFNPT RVF MÎN f (x) = 4 EPOEF f (x) = √ x→4 x−2 EPNJOJP EF f FT FM DPOKVOUP BCJFSUP (0, 4) ∪ (4, +∞) Z  FT VO QVOUP EF BDVNVMBDJÓO EF EJDIP DPOKVOUP /PUFNPT RVF   √  (x − 4) ( x + 2)  √  √ |f (x) − 4| =  √ − 4 = | x − 2|. ( x − 2) ( x + 2) √ √ "EFNÃT |x − 4| = | x + 2| | x − 2| OPT NPUJWB B FODPOUSBS VOB DPUB √ JOGFSJPS EF x + 2 RVF QPEFNPT IBMMBS EF MB TJHVJFOUF NBOFSB TJ δ < 1 Z |x − 4| < δ FOUPODFT |x − 4| < 1 Z BTÎ 3 < x < 5 FO QBSUJDVMBS √ x > 1 -VFHP x + 2 > 3 %F FTUP TF EFEVDF RVF TJ δ = NÎO {1, 3ε} Z |x − 4| < δ FOUPODFT √ δ |x − 4| < ≤ ε. |f (x) − 4| = | x − 2| = √ 3 x+2 

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-BT TVDFTJPOFT VOB WF[ NÃT EFNVFTUSBO TFS VOB IFSSBNJFOUB QP EFSPTB &M QSÓYJNP UFPSFNB QSPQPSDJPOB VO DSJUFSJP QPS TVDFTJPOFT QBSB EFUFSNJOBS TJ VO OÙNFSP SFBM L FT MÎNJUF P OP EF VOB GVODJÓO 5FPSFNB  $SJUFSJP QBSB MÎNJUFT QPS TVDFTJPOFT  4FBO f : A ⊆ R −→ R Z c ∈ R VO QVOUP EF BDVNVMBDJÓO EF A &OUPODFT MÎN f (x) = L ⇐⇒ (∀ {xn }) (xn ∈ A  {c} ∧ xn → c ⇒ f (xn ) → L).

x→c

%FNPTUSBDJÓO 4VQPOHBNPT RVF MÎN f (x) = L Z TFB {xn } VOB TVDFTJÓO x→c EF QVOUPT FO A UBM RVF xn = c QBSB UPEP n ∈ N Z xn → c. 1SPCFNPT RVF f (xn ) → L. 4FB QVFT ε > 0 $PNP MÎN f (x) = L FYJTUF δ > 0 UBM x→c

RVF TJ x ∈ A Z 0 < |x − c| < δ FOUPODFT |f (x) − L| < ε. "IPSB CJFO ZB RVF xn → c FYJTUF N0 ∈ N UBM RVF TJ n ≥ N0 FOUPODFT 0 < |xn − c| < δ Z QPS UBOUP QBSB UPEP n ≥ N0 TF UJFOF RVF |f (xn ) − L| < ε MP DVBM TJHOJàDB RVF f (xn ) → L 1BSB QSPCBS MB PUSB JNQMJDBDJÓO FT NÃT TFODJMMP QPS SFEVDDJÓO BM BC TVSEP TVQPOHBNPT RVF OP TF UJFOF MB DPODMVTJÓO FT EFDJS MÎN f (x) = L x→c &OUPODFT FYJTUF ε0 > 0 UBM RVF QBSB UPEP δ > 0 FYJTUF x ∈ A, x = c RVF EFQFOEF EF δ DPO 0 < |x − c| < δ Z |f (x) − L| ≥ ε0  6TBOEP FTUP DPO δ = 1/n EPOEF n ≥ 1 FT OBUVSBM FODPOUSBNPT VOB TVDFTJÓO xn ∈ A  {c} UBM RVF xn → c Z |f (xn ) − L| ≥ ε0 MP RVF DPOUSBEJDF MB  IJQÓUFTJT %F FTUB NBOFSB RVFEB QSPCBEP FM UFPSFNB  "QSPWFDIBOEP FTUB DBSBDUFSJ[BDJÓO QPS TVDFTJPOFT Z MBT QSPQJFEBEFT EF FTUBT QPEFNPT EFNPTUSBS MBT QSPQJFEBEFT CÃTJDBT EF MPT MÎNJUFT 5FPSFNB  4FBO f, g : A ⊆ R −→ R GVODJPOFT Z c ∈ R VO QVOUP EF BDVNVMBDJÓO EF A 4J MÎN f (x) = L Z MÎN g(x) = M

x→c

x→c

  Z TJ K FT VOB DPOTUBOUF SFBM FOUPODFT MÎN Kf (x) MÎN f (x) + g(x) x→c x→c   Z MÎN f (x) · g(x) FYJTUFO Z BEFNÃT x→c

    MÎN Kf (x) = KL, MÎN f (x)+g(x) = L+M Z MÎN f (x)·g(x) = L·M.

x→c

x→c

x→c

  4J BEFNÃT g(x) = 0 QBSB UPEP x ∈ A Z M = 0 FOUPODFT MÎN f (x)/g(x) FYJTUF Z FT JHVBM B L/M.



x→c

-PT OÙNFSPT SFBMFT

%FNPTUSBDJÓO 4FB {xn } VOB TVDFTJÓO EF QVOUPT FO A UBM RVF xn = c QBSB UPEP n ∈ N Z xn → c. 1PS MB IJQÓUFTJT f (xn ) → L Z g(xn ) → M DVBOEP n → ∞. 1PS QSPQJFEBEFT EF MBT TVDFTJPOFT DPO WFSHFOUFT DPODMVJNPT RVF Kf (xn ) → KL, f (xn ) + g(xn ) → L + M Z f (xn ) · g(xn ) → LM EF MP RVF TF TJHVF RVF Kf (x) → KL, f (x) + g(x) → L + M Z f (x) · g(x) → L · M DVBOEP x → c. 'JOBMNFOUF WFBNPT RVF f (x)/g(x) → L/M DVBOEP x → c : TFB ε > 0. 1PS MB QSJNFSB DPODMVTJÓO EF FTUF UFPSFNB f (xn )M − g(xn )L → 0 DVBOEP n → ∞ Z BTÎ FYJTUF N ∈ N UBM RVF |f (xn )M − g(xn )L| < ε QBSB UPEP n ≥ N  1PS PUSB QBSUF FYJTUF N1 ∈ N UBM RVF QBSB UPEP n ≥ N1 UFOFNPT RVF |g(xn ) − M | < |M |/2 Z EF FTUP ÙMUJNP TF EFEVDF RVF |g(xn )| > |M |/2 QBSB UPEP n ≥ N1  -VFHP TJ N0 = NÃY{N, N1 } Z UFOJFOEP FO DVFOUB MBT DPODMVTJPOFT QSFWJBT PCUFOFNPT RVF QBSB UPEP n ≥ N0    f (xn )  |f (xn )M − g(xn )L| L 2ε   < 2·  g(xn ) − M  = |g(xn ) M | M %F FTUB NBOFSB RVFEB EFNPTUSBEP FM UFPSFNB

 

"DUP TFHVJEP OPT PDVQBSFNPT EF MB OPDJÓO EF DPOUJOVJEBE *OUVJUJ WBNFOUF DPOUJOVJEBE OPT JOEJDB iBVTFODJB EF TBMUPTu $POTJEFSBSFNPT GVODJPOFT EFàOJEBT FO VO TVCDPOKVOUP EF MPT OÙNFSPT SFBMFT Z RVF UPNBO WBMPSFT SFBMFT &O QBMBCSBT TJNQMFT f FT DPOUJOVB FO VO QVOUP EF TV EPNJOJP x0 TJ FT QPTJCMF MPHSBS RVF f (x) FTUÊ iBSCJ USBSJBNFOUFu DFSDB EF f (x0 ) DVBOEP x FTUÃ iTVàDJFOUFNFOUF QSÓYJNP B x0 u %FàOJDJÓO  4FBO A ⊆ R f : A → R VOB GVODJÓO Z x0 ∈ A %FDJNPT RVF f FT DPOUJOVB FO x0 TJ TF DVNQMF MP TJHVJFOUF ∀ε > 0 ∃ δ > 0 UBM RVF TJ x ∈ A DPO |x − x0 | < δ FOUPODFT |f (x) − f (x0 )| < ε. 0CTFSWBDJPOFT J &O HFOFSBM FM OÙNFSP δ EFQFOEF EF f, ε Z x0  JJ -B BOUFSJPS EFàOJDJÓO QVFEF GPSNVMBSTF DMBSBNFOUF FO UÊSNJOPT EF CPMBT BCJFSUBT MB GVODJÓO f FT DPOUJOVB FO x0 ∈ A TJ ∀ ε > 0 ∃ δ > 0 : f [#δ (x0 )] ⊆ #ε (f (x0 ));





5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

FT EFDJS TJ EBEB DVBMRVJFS εCPMB BCJFSUB FO R DPO DFOUSP f (x0 ) FYJTUF VOB δCPMB BCJFSUB FO A EF DFOUSP x0 DVZB JNBHFO CBKP f FTUÃ DPOUFOJEB FO MB CPMB JOJDJBM JJJ -B OPDJÓO EF DPOUJOVJEBE QVFEF TFS GPSNVMBEB VUJMJ[BOEP FM DPODFQUP EF DPOKVOUP BCJFSUP MB GVODJÓO f FT DPOUJOVB FO x0 TJ Z TPMP TJ QBSB DVBMRVJFS TVCDPOKVOUP BCJFSUP G ⊆ R RVF DPOUFOHB B f (x0 ) FYJTUF VO TVCDPOKVOUP BCJFSUP G EF R RVF DPOUFOHB B x0 Z f [G] ⊂ G  &O FGFDUP TFB f DPOUJOVB FO x0 Z TFB G ⊂ R VO BCJFSUP UBM RVF f (x0 ) ∈ G  %FCJEP B RVF G FT BCJFSUP FYJTUF ε0 > 0 UBM RVF #ε0 (f (x0 )) ⊂ G Z QPS MB DPOUJOVJEBE EF f FO x0 FYJTUF δ0 > 0 UBM RVF f [#δ (x0 )] ⊆ #ε0 (f (x0 )) 5PNBOEP G = #δ (x0 ) TF UJFOF RVF G FT VO BCJFSUP FO R UBM RVF x0 ∈ G Z f [G] ⊂ G  3FDÎQSPDBNFOUF TVQPOHBNPT RVF QBSB UPEP BCJFSUP G FO R UBM RVF f (x0 ) ∈ G FYJTUF VO BCJFSUP G FO R UBM RVF x0 ∈ G Z f [G] ⊂ G  7FBNPT RVF f FT DPOUJOVB FO x0  4FB ε > 0 EBEP $PNP G = #ε (f (x0 )) FT VO BCJFSUP FO R RVF DPOUJFOF B f (x0 ) FYJTUF G ⊂ R G BCJFSUP x0 ∈ G Z f [G] ⊂ G  $PNP G FT BCJFSUP Z x0 ∈ G FYJTUF δ > 0 UBM RVF #δ (x0 ) ⊂ G -VFHP f [#δ (x0 )] ⊂ f [G] ⊂ G = #ε (f (x0 )); JF G FT DPOUJOVB FO x0  JW -B DPOUJOVJEBE QVOUVBM QVFEF GPSNVMBSTF FO UÊSNJOPT EF TVDFTJPOFT f FT DPOUJOVB FO VO QVOUP c EF TV EPNJOJP TJ Z TPMP TJ QBSB UPEB TVDFTJÓO {xn } FO FM EPNJOJP EF f UBM RVF xn → c TF DVNQMF RVF f (xn ) → f (c) -B WFSJàDBDJÓO EF FTUB BàSNBDJÓO FT DPNP FO FM 5FPSFNB  &O PUSPT UÊSNJOPT FTUBNPT EJDJFOEP RVF f FT DPOUJOVB FO c ⇐⇒ (∀ {xn } ⊂ %PN f ) % & xn → c ⇒ MÎN f (xn ) = f ( MÎN xn ) . n→∞

n→∞

%FàOJDJÓO  %FDJNPT RVF VOB GVODJÓO f : A ⊆ R → R FT DPOUJOVB TJ QBSB UPEP x ∈ A G FT DPOUJOVB FO x &KFNQMP  $POTJEFSFNPT MB GVODJÓO f : R → R EFàOJEB QPS f (x) = x2 QBSB UPEP x ∈ R 7FBNPT RVF f FT DPOUJOVB FO R 4FB x0 ∈ R Z EFNPTUSFNPT RVF QBSB DVBMRVJFS ε > 0 QPEFNPT FODPOUSBS VO δ > 0 UBM RVF |x − x0 | < δ ⇒ |x2 − x20 | < ε &O FGFDUP |x2 − x20 | = |(x + x0 )(x − x0 )| = |(x − x0 + 2x0 )(x − x0 )| ≤ (|x − x0 | + 2|x0 |)|x − x0 |.



-PT OÙNFSPT SFBMFT

1PS MP UBOUP TJ 0 < δ ≤ 1 FOUPODFT |x − x0 | < δ ⇒ |x2 − x20 | ≤ (1 + 2|x0 |)|x − x0 |. 4J BEFNÃT δ ≤

ε 1+2|x0 |

UFOESFNPT RVF

|x − x0 | < δ ⇒ |x2 − x20 | < ε. ε } TF UJFOF RVF %F FTUB NBOFSB TJ δ ≤ NÎO {1, 1+2|x 0|

|x − x0 | < δ ⇒ |x2 − x20 | < ε. -VFHP f (x) = x2 FT DPOUJOVB FO x0 Z DPNP FTUF FT BSCJUSBSJP FOUPODFT f FT DPOUJOVB FO UPEP R. &KFNQMP  -B GVODJÓO f : R → R EFàOJEB QPS

0 si x < 0 f (x) = 1 si x ≥ 0 FT DPOUJOVB QBSB UPEP x = 0 QFSP OP FT DPOUJOVB FO x = 0 -B QSJNFSB QBSUF FT JONFEJBUB QVFT f FT DPOTUBOUF FO DBEB JOUFSWBMP (−∞, 0) Z (0, +∞) %FNPTUSFNPT BIPSB RVF f OP FT DPOUJOVB FO x0 = 0 1BSB ε = 12 > 0 OP FYJTUF δ > 0 UBM RVF 1 |x| < δ ⇒ |f (x) − f (0)| < · 2 &O FGFDUP TJ FYJTUJFSB UBM δ > 0 FO QBSUJDVMBS QBSB x ∈ (−δ, 0) UFOESÎB NPT |x| < δ Z TJO FNCBSHP |f (x) − f (0)| = |0 − 1| = 1 > 12  &YJTUFO GVODJPOFT RVF TPO EJTDPOUJOVBT FO UPEP QVOUP EF TV EPNJOJP &KFNQMP  -B GVODJÓO EF %JSJDIMFU f : R → R EFàOJEB QPS

1 si x es racional f (x) = 0 si x es irracional, FT EJTDPOUJOVB FO UPEP QVOUP 4FB x0 VO OÙNFSP SBDJPOBM WFBNPT RVF f OP FT DPOUJOVB FO x0  &O FGFDUP QBSB ε0 = 12 > 0 TF DVNQMF MB OFHBDJÓO EF   EBEP δ > 0

 f [#δ (x0 )] = f [(x0 − δ, x0 + δ)] ⊆ #ε0 (f (x0 )) = #1/2 (1) , ZB RVF ∀ δ > 0 FYJTUF y0 ∈ #(x0 , δ) UBM RVF y0 FT JSSBDJPOBM Z BTÎ 0 = f (y0 ) ∈ f [(x0 − δ, x0 + δ)] / ( 12 , 32 ) QFSP 0 = f (y0 ) ∈ %F NBOFSB TJNJMBS TF QSVFCB DPO FM NJTNP ε0 = JSSBDJPOBM f OP FT DPOUJOVB FO x0 

1 2

> 0 RVF TJ x0 FT VO



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP  "OBMJDFNPT MB DPOUJOVJEBE EF MB GVODJÓO f : R → R EFàOJEB QPS

0 si x es irracional f (x) = p 1 q si x = q , EPOEF p Z q TPO QSJNPT SFMBUJWPT Z q > 0 $POTJEFSFNPT f (0) = 1. "àSNBNPT RVF f OP FT DPOUJOVB FO Q 4VQPOHBNPT x0 ∈ Q DPO x0 = pq EPOEF p Z q TPO QSJNPT SFMBUJWPT Z q > 0 1PS EFàOJDJÓO f (x0 ) = 1q  1 1BSB ε0 = 2q > 0 OP FYJTUF δ > 0 UBM RVF x ∈ R ∧ |x − x0 | < δ ⇒ |f (x) − f (x0 )| <

1 · 2q

&O FGFDUP ∀ δ > 0 ∃ i ∈ I UBM RVF |i − x0 | < δ QFSP |f (i) − f (x0 )| =

1 1 > ε0 = . q 2q

-VFHP f OP FT DPOUJOVB FO x0  "IPSB WFBNPT RVF f FT DPOUJOVB FO I. 4FBO i0 ∈ I Z ε > 0 1PS MB QSP QJFEBE BSRVJNFEJBOB FYJTUF N ∈ N UBM RVF N1 < ε 1BSB j ∈ {1, . . . , N } m m +1 FYJTUF mj := [[ j i0 ]] ∈ Z UBM RVF jj < i0 < jj · 4FBO δj = NÎO {i0 −

mj mj + 1 , − i0 } > 0, j = 1, . . . , N j j

Z δ = NÎO {δ1 , . . . , δN } > 0 /PUFNPT RVF DPNP MPT FOUFSPT mj Z mj + 1 m m +1 TPO MPT ÓQUJNPT RVF DPOUJFOFO B j i0 FOUPODFT jj Z jj TPO MPT SBDJP OBMFT ÓQUJNPT RVF DPOUJFOFO B i0  1PS UBOUP FO UÊSNJOPT HFPNÊUSJDPT FM OÙNFSP δ FT MB NÎOJNB EJTUBODJB FOUSF FM OÙNFSP JSSBDJPOBM i0 Z MPT m m +1 SBDJPOBMFT jj Z jj MPT DVBMFT UJFOFO EFOPNJOBEPS NFOPS P JHVBM RVF N  %FNPTUSFNPT RVF x ∈ R ∧ |x − i0 | < δ ⇒ |f (x) − f (i0 )| = |f (x)| < ε. &GFDUJWBNFOUF TJ x ∈ I Z |x − i0 | < δ FOUPODFT |f (x) − f (i0 )| = 0 < ε. "IPSB TVQPOHBNPT RVF x ∈ Q DPO x = nk EPOEF k Z n TPO QSJNPT SFMBUJWPT n > 0 Z |x − i0 | < δ "àSNBNPT RVF OFDFTBSJBNFOUF n > N DPNP FT EF FTQFSBSTF HFPNÊUSJDBNFOUF Z BTÎ VOB WF[ KVTUJàDBEP FTUF IFDIP TF UJFOF RVF |f (x) − f (i0 )| =



1 1 < < ε, n N

-PT OÙNFSPT SFBMFT

DPO MP RVF TF QSVFCB MB DPOUJOVJEBE EF f FO i0 ∈ I %FNPTUSFNPT MB ÙMUJNB BàSNBDJÓO QVFTUP RVF |x − i0 | = | nk − i0 | < δ ≤ δj QBSB UPEP j ∈ {1, . . . , N } FOUPODFT mj + 1 mj k < < , j n j

QBSB UPEP j ∈ {1, . . . , N }.



4J PDVSSJFSB RVF n ≤ N IBDJFOEP j = n FO  FODPOUSBSÎBNPT VO OÙNFSP FOUFSP FOUSF EPT FOUFSPT DPOTFDVUJWPT MP RVF QPS TVQVFTUP FT VO BCTVSEP &TUP QSVFCB OVFTUSB BàSNBDJÓO FT EFDJS n > N. &KFSDJDJP  4FB f : R −→ R VOB GVODJÓO DPOUJOVB FO  Z UBM RVF f (x) = f (2x) QBSB UPEP x ∈ R 1SVFCF RVF f FT DPOTUBOUF &KFSDJDJP  1SVFCF FM TJHVJFOUF UFPSFNB TFB f : R −→ R VOB GVODJÓO &OUPODFT f FT DPOUJOVB TJ Z TPMP TJ QBSB UPEP BCJFSUP O FO R, f −1 (O) FT BCJFSUP FO R TJ Z TPMP TJ QBSB UPEP DFSSBEP F FO R, f −1 (F) FT DFSSBEP FO R. &KFSDJDJP  4FB f : R −→ R VOB GVODJÓO DPOUJOVB %FNVFTUSF RVF QBSB UPEP E ⊆ R TF DVNQMF RVF f (E) ⊆ f (E). 0CTFSWBDJÓO 5PEB GVODJÓO FT DPOUJOVB FO VO QVOUP BJTMBEP EF TV EP NJOJP TFBO A FM EPNJOJP EF VOB GVODJÓO EF WBMPS SFBM f, x0 VO QVOUP BJTMBEP EF A Z ε > 0 $PNP x0 FT VO QVOUP BJTMBEP EF A FOUPODFT FYJTUF δ > 0 UBM RVF A ∩ #δ (x0 ) = {x0 } Z BTÎ QBSB UPEP x ∈ A DPO |x − x0 | < δ FM ÙOJDP FT x = x0

TF UJFOF RVF |f (x) − f (x0 )| = 0 < ε. /ÓUFTF RVF FM OÙNFSP δ WJFOF EBEP QPS MB EFàOJDJÓO EF QVOUP BJTMBEP Z QPS UBOUP OP EFQFOEF EF ε $PO MP FYQVFTUP UJFOF TFOUJEP FYBNJOBS MB DPOUJOVJEBE QVOUVBM EF VOB GVODJÓO TPMBNFOUF FO MPT QVOUPT EF BDVNVMBDJÓO EF TV EPNJOJP Z FO DPOTFDVFODJB QVFEF TFS WFSJàDBEB WÎB MÎNJUFT DPNP TF FTUBCMFDF FO FM TJHVJFOUF SFTVMUBEP DVZB EFNPTUSBDJÓO FT VOB BQMJDBDJÓO EJSFDUB EF MBT EFàOJDJPOFT 5FPSFNB  4FBO A ⊆ R f : A → R VOB GVODJÓO Z x0 ∈ A  &OUPODFT f FT DPOUJOVB FO x0 TJ Z TPMP TJ MÎN f (x) = f (x0 ). x→x0

&KFSDJDJP  4FB f : R −→ R DPOUJOVB FO  Z BEJUJWB FTUP FT QBSB UPEP x, y ∈ R TF DVNQMF f (x + y) = f (x) + f (y). %FNVFTUSF RVF f FT DPOUJOVB FO UPEP QVOUP &KFSDJDJP  )BMMF UPEBT MBT GVODJPOFT DPOUJOVBT f : R −→ R UBMFT RVF f (a + b) = f (ab) QBSB UPEP a, b ∈ I.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-B DPOUJOVJEBE FT QSFTFSWBEB QPS MBT PQFSBDJPOFT BMHFCSBJDBT EF GVO DJPOFT DPOUJOVBT "OUFT EF QSPCBS FTUF IFDIP EFNPTUSBNPT VO MFNB JNQPSUBOUF RVF TFSÃ EF VUJMJEBE FO VOB EF MBT BàSNBDJPOFT &M MFNB HB SBOUJ[B RVF VOB GVODJÓO DPOUJOVB FO VO QVOUP QSFTFSWB FM TJHOP FO VOB WFDJOEBE EFM QVOUP -FNB  4FB f VOB GVODJÓO DPOUJOVB FO VO QVOUP c EF TV EPNJOJP Z UBM RVF f (c) > 0 &OUPODFT FYJTUF VO δCPMB #δ (c) EF c UBM RVF f (x) > 0 QBSB UPEP x ∈ #δ (c). 3FTVMUBEP TJNJMBS TF UJFOF TJ f (c) < 0 %FNPTUSBDJÓO $PNP f FT DPOUJOVB FO c FOUPODFT QBSB ε = 12 f (c) > 0 FYJTUF δ > 0 UBM RVF TJ x FTUÃ FO FM EPNJOJP EF f ≡ A Z |x−c| < δ TF UJFOF |f (x) − f (c)| < 12 f (c) %F FTUP TF TJHVF RVF 0 < 12 f (c) < f (x) < 32 f (c) QBSB UPEP x ∈ A DPO |x − c| < δ &O FM DBTP f (c) < 0 TF SB[POB BOÃMPHBNFOUF DPO ε = − 12 f (c) > 0 QBSB DPODMVJS RVF 3 1 f (c) < f (x) < f (c) < 0 2 2 QBSB UPEP x ∈ A Z FO DJFSUB WFDJOEBE EF c -B QSVFCB EFM MFNB RVFEB  DPNQMFUB  0CTFSWBDJÓO 4J VOB GVODJÓO f FT DPOUJOVB FO VO QVOUP c EPOEF f (c) = 0 OP OFDFTBSJBNFOUF TF QVFEF BàSNBS RVF MB GVODJÓO f TF BOVMB FO BMHVOB WFDJOEBE EF c DPNP QVFEF TFS JMVTUSBEP DPO f (x) = x Z c = 0 P VO DBTP NÃT FYÓUJDP FT FM &KFNQMP  5FPSFNB  4FBO f Z g GVODJPOFT DPOUJOVBT FO VO QVOUP c Z K VOB DPOTUBOUF SFBM &OUPODFT MBT GVODJPOFT Kf, f + g Z f g TPO DPOUJOVBT FO c 4J BEFNÃT g(c) = 0 FOUPODFT f /g FT DPOUJOVB FO FM QVOUP c. %FNPTUSBDJÓO 6TBNPT VO BSHVNFOUP QPS TVDFTJPOFT TFB {xn } VOB TVDFTJÓO UBM RVF xn → c $PNP f Z g TPO DPOUJOVBT FO c FOUPODFT f (xn ) → f (c) Z g(xn ) → g(c) 1SPQJFEBEFT EF MBT TVDFTJPOFT DPOWFS HFOUFT JNQMJDBO RVF Kf (xn ) → Kf (c) Z f (xn ) + g(xn ) → f (c) + g(c), f (xn ) · g(xn ) → f (c) · g(c), MP DVBM TJHOJàDB RVF Kf, f + g Z f g TPO DPOUJOVBT FO c 'JOBMNFOUF QPS FM MFNB QSFWJP FYJTUF δ > 0 UBM RVF g(x) = 0 QBSB UPEP x ∈ #δ (c) -VFHP DPNP xn → c FYJTUF N0 ∈ N UBM RVF xn ∈ #δ (c) QBSB UPEP n ≥ N0 Z QPS UBOUP g(xn ) = 0 QBSB FTUB N0 DPMB 1PEFNPT FOUPODFT BQMJDBS MB QSPQJFEBE EF FYJTUFODJB EFM MÎNJUF EF VO DPDJFOUF EF TVDFTJPOFT



-PT OÙNFSPT SFBMFT

DPOWFSHFOUFT DVBOEP MB TVDFTJÓO EFM EFOPNJOBEPS UJFOF DPMB OP OVMB Z DPOWFSHF B VO SFBM EJTUJOUP EF DFSP QBSB DPODMVJS RVF f (xn )/g(xn ) → f (c)/g(c). &TUP EFNVFTUSB MB DPOUJOVJEBE EF f /g FO FM QVOUP c.

 

1PS JOEVDDJÓO TF QVFEF QSPCBS RVF VOB TVNB àOJUB EF GVODJPOFT DPOUJOVBT UBNCJÊO FT DPOUJOVB 4JNJMBSNFOUF QBSB VO QSPEVDUP EF VO OÙNFSP àOJUP EF GVODJPOFT DPOUJOVBT &KFSDJDJP  -BT TJHVJFOUFT BTFWFSBDJPOFT TF QVFEFO DPOTJEFSBS UFP SFNBT B 4FBO a < b OÙNFSPT SFBMFT Z f, g : [a, b] −→ R GVODJPOFT DPOUJ OVBT 1SVFCF RVF FM DPOKVOUP {x ∈ [a, b] : f (x) = g(x)} FT DFSSBEP C #BKP MBT NJTNBT DPOEJDJPOFT EF B QSVFCF RVF FM DPOKVOUP {x ∈ [a, b] : f (x) < g(x)} FT BCJFSUP D 4VQPOHB RVF f : R −→ R FT DPOUJOVB .VFTUSF RVF QBSB UPEP α ∈ R FM DPOKVOUP {x ∈ [a, b] : f (x) ≤ α} FT DFSSBEP Z RVF {x ∈ [a, b] : f (x) < α} FT VO DPOKVOUP DFSSBEP &KFSDJDJP  4FBO f Z g GVODJPOFT EF WBMPS SFBM Z DPOUJOVBT FO FM DPOKVOUP [a, b] 1SVFCF RVF FM DPOKVOUP K = {x ∈ [a, b] : f (x) ≥ g(x)} FT DPNQBDUP -B DPOUJOVJEBE UBNCJÊO FT QSFTFSWBEB QPS MB DPNQPTJDJÓO EF GVO DJPOFT DPOUJOVBT 5FPSFNB  4FBO f : A −→ R Z g : B −→ R GVODJPOFT UBMFT RVF f (A) ⊆ B 4J f FT DPOUJOVB FO c ∈ A Z g FT DPOUJOVB FO f (c) ∈ B FOUPODFT g ◦ f : A −→ R FT DPOUJOVB FO c %FNPTUSBDJÓO 4FB ε > 0 EBEP $PNP g FT DPOUJOVB FO f (c) FYJTUF η > 0 UBM RVF y ∈ B ∧ |y − f (c)| < η ⇒ |g(y) − g(f (c)| < ε.



1PS PUSB QBSUF QPS MB DPOUJOVJEBE EF f FO c QBSB FTUF η > 0 FYJTUF δ > 0 UBM RVF x ∈ A ∧ |x − c| < δ ⇒ |f (x) − f (c)| < η. 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

1PS DPOTJHVJFOUF DPNCJOBOEP  Z 

UFOFNPT RVF x ∈ A ∧ |x − c| < δ ⇒ |g(f (x)) − g(f (c))| < ε, FT EFDJS g ◦ f FT DPOUJOVB FO c

 

%FàOJDJÓO  6OB GVODJÓO f : A −→ R TF EJDF BDPUBEB FO A TJ TV SBOHP f (A) FT VO DPOKVOUP BDPUBEP FO R FT EFDJS TJ FYJTUF VOB DPOTUBOUF C > 0 JOEFQFOEJFOUF EF x UBM RVF ∀ x ∈ A, |f (x)| ≤ C. -B DPOTUBOUF C TÎ EFQFOEF EFM EPNJOJP FO FM TFOUJEP EF RVF BM DBN CJBS FM EPNJOJP EF MB GVODJÓO MB DPOTUBOUF DBNCJB QFSP VOB WF[ àKBEP FM EPNJOJP MB DPOTUBOUF OP EFQFOEF EF x FO FM EPNJOJP EF MB GVODJÓO "MHVOPT FKFNQMPT JMVTUSBO FTUP DMBSBNFOUF &KFNQMP  -B GVODJÓO f (x) = 1/x FT BDPUBEB FO A = [1, +∞) ZB RVF QBSB UPEP x ∈ A, x ≥ 1 Z BTÎ |f (x)| = x1 ≤ 1. 4J DBNCJBNPT FM EPNJOJP EJHBNPT A = (1/4, +∞) FOUPODFT MB DPOTUBOUF DBNCJB QVFT QBSB UPEP x ∈ A, |f (x)| < 4 "IPSB TJ FTDPHFNPT FM EPNJOJP A = (0, 1) 1 DPO n ≥ 1 OBUVSBM ZB MB GVODJÓO f OP FT BDPUBEB BMMÎ QVFT QBSB xn = n+1 TF UJFOF RVF |f (xn )| = n + 1 MP RVF UJFOEF B +∞ 0CTFSWBDJÓO &T JNQPSUBOUF NFODJPOBS RVF TJ VOB GVODJÓO f : A −→ R OP FT BDPUBEB FO A FOUPODFT FYJTUF VOB TVDFTJÓO {xn } EF QVOUPT FO A UBM RVF |f (xn )| → +∞ FGFDUJWBNFOUF MB OFHBDJÓO EF GVODJÓO BDPUBEB FRVJWBMF B EFDJS RVF ∀ C > 0 ∃ x ∈ A : |f (x)| > C. "QMJDBOEP FTUP DPO C = n ∈ N FODPOUSBNPT xn ∈ A UBM RVF |f (xn )| > n. &TUP KVTUJàDB OVFTUSB BàSNBDJÓO &KFNQMP  4FBO A ∈ R, b ∈ R Z m ∈ N∗ DPOTUBOUFT àKBT 4J f : (b − 1, b + 1) −→ R FTUÃ EBEB QPS f (x) = Axm FOUPODFT f FT BDPUBEB FO (b − 1, b + 1) ZB RVF QBSB UPEP x FO FTUF EPNJOJP TF DVNQMF RVF |x| ≤ |x − b| + |b| < 1 + |b| Z QPS DPOTJHVJFOUF |f (x)| ≤ A(1 + |b|)m , FT EFDJS FYJTUF VOB DPOTUBOUF C := A(1 + |b|)m UBM RVF QBSB UPEP x ∈ (b − 1, b + 1) TF DVNQMF RVF |f (x) ≤ C 7BNPT B BQSPWFDIBS FTUP QBSB EFNPTUSBS RVF UPEP QPMJOPNJP DPO DPFàDJFOUFT SFBMFT EFàOF VOB GVODJÓO DPOUJOVB WFS MB TFDDJÓO " QÃH  



-PT OÙNFSPT SFBMFT

4FB p : R −→ R EFàOJEB QPS p(x) = an xn + an−1 xn−1 + · · · + a0 EPOEF UBOUP n ∈ N∗ DPNP DBEB DPFàDJFOUF ai ∈ R FTUÃO àKBEPT Z an = 0 "àSNBNPT RVF p FT VOB GVODJÓO DPOUJOVB 1BSB FMMP UPNFNPT b ∈ R BSCJUSBSJP Z MP EFKBNPT àKP FO FM SB[POBNJFOUP RVF TJHVF 1PS FM BMHPSJUNP EF MB EJWJTJÓO QBSB QPMJOPNJPT Z FM UFPSFNB EFM SFTJEVP WFS 5FPSFNBT " Z " QÃH  FYJTUF VO ÙOJDP QPMJOPNJP c EF HSBEP n − 1 MMBNBEP FM DPDJFOUF Z UBM RVF p(x) = c(x)(x − b) + p(b) EPOEF c(x) = An−1 xn−1 + An−2 xn−2 + · · · + A1 x + A0 . 4FB ε > 0 Z IBMMFNPT δ > 0 UBM RVF TJ x ∈ R Z |x − b| < δ FOUPODFT |p(x) − p(b)| < ε $PNP |p(x) − p(b)| = |c(x)| |x − b| CBTUB RVF BDP UFNPT |c(x)| 0CTFSWFNPT RVF DBEB UÊSNJOP RVF EFàOF BM QPMJOPNJP c FT EF MB GPSNB f (x) = Axm Z TJ SFTUSJOHJNPT δ EF NBOFSB RVF δ ≤ 1 FOUPODFT QPS MP IFDIP BM JOJDJP TBCFNPT RVF FYJTUFO DPOTUBOUFT QPTJUJWBT C1 , C2 , · · · , Cn−1 UBMFT RVF TJ |x − b| < 1 FOUPODFT |c(x)| ≤ Cn−1 + Cn−2 + · · · C1 + A0 ≡ C. -VFHP TJ δ = NÎO {1, ε/2C} Z |x − b| < δ FOUPODFT |p(x) − p(b)| < ε MP RVF EFNVFTUSB MB DPOUJOVJEBE EFM QPMJOPNJP p FO b Z DPNP FTUPT TPO BSCJUSBSJPT DPODMVJNPT RVF UPEP QPMJOPNJP FT DPOUJOVP FO R -B JEFB EF QSPCBS MB DPOUJOVJEBE EF VO QPMJOPNJP DPNP TF IB IFDIP TF BQPZB FO iPCMJHBSu BM MFDUPS B SFDPSEBS P SFWJTBS BMHVOPT UFPSFNBT JNQPSUBOUFT EF QPMJOPNJPT DPNP FM BMHPSJUNP EF MB EJWJTJÓO FM UFPSFNB EFM SFTJEVP Z PUSPT IFDIPT RVF TF EFTDSJCFO FO FM BQÊOEJDF -B QSVFCB QVEP TFS QSFTFOUBEB EF PUSB GPSNB DBEB UÊSNJOP RVF EFàOF BM QPMJOPNJP p FT EF MB GPSNB Axm FM DVBM FT VO QSPEVDUP EF m + 1 GBDUPSFT Z DBEB VOP EF FMMPT FT DPOUJOVP -VFHP FM QPMJOPNJP p SFTVMUB DPOUJOVP QVFT FT MB TVNB EF VO OÙNFSP àOJUP EF FTUPT UÊSNJOPT -B DPOUJOVJEBE EF VOB GVODJÓO f OP FT TVàDJFOUF QBSB RVF FTUB TFB BDPUBEB &M &KFNQMP  NVFTUSB VOB GVODJÓO DPOUJOVB RVF FT BDPUBEB FO VO DPOKVOUP DFSSBEP OP BDPUBEP Z UBNCJÊO FT BDPUBEB FO VO BCJFSUP OP BDPUBEP QFSP OP MP FT FO VO DPOKVOUP BCJFSUP BDPUBEP &M TJHVJFOUF SFTVMUBEP EJDF RVF BEFNÃT EF DPOUJOVJEBE EF MB GVODJÓO TF SFRVJFSF RVF FM EPNJOJP TFB DPNQBDUP DFSSBEP Z BDPUBEP  5FPSFNB  4FBO K ⊂ R DPNQBDUP Z f : K −→ R VOB GVODJÓO DPOUJOVB &OUPODFT f FT BDPUBEB %FNPTUSBDJÓO 4VQPOHBNPT RVF f OP FT BDPUBEB -VFHP FYJTUF VOB TV DFTJÓO {xn } ⊂ K UBM RVF QBSB UPEP n ∈ N TF DVNQMF RVF |f (xn )| > n



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

$PNP K FT BDPUBEP FOUPODFT MB TVDFTJÓO {xn } FT BDPUBEB Z QPS UBOUP FM UFPSFNB EF #PM[BOP8FJFSTUSBTT JNQMJDB RVF FYJTUF VOB TVCTVDFTJÓO {xnk } RVF DPOWFSHF FT EFDJS FYJTUF a ∈ R UBM RVF xnk → a DVBOEP k → ∞ %FCJEP B RVF K FT DFSSBEP a ∈ K Z QPS MB DPOUJOVJEBE EF f PCUFOFNPT RVF f (xnk ) → f (a) DVBOEP k → ∞ MP RVF QFSNJUF BàSNBS RVF {f (xnk )} FT BDPUBEB 1FSP FM TVQVFTUP RVF FTUBNPT IBDJFOEP EJDF RVF |f (xnk )| > nk ≥ k QBSB UPEP k ∈ N MP DVBM DPOUSBEJDF FM BDPUB  NJFOUP EF {f (xnk )} &TUB DPOUSBEJDDJÓO QSVFCB RVF f FT BDPUBEB  &M UFPSFNB BOUFSJPS HBSBOUJ[B RVF TJ f FT DPOUJOVB FO FM DPNQBDUP K FOUPODFT MPT OÙNFSPT ÎOG f Z TVQ f FYJTUFO &O SFBMJEBE TF BMDBO[BO K

K

MPT FYUSFNPT NÃT QSFDJTBNFOUF UFOFNPT 5FPSFNB  5FPSFNB EFM WBMPS FYUSFNP  4FB f VOB GVODJÓO EF WBMPS SFBM Z DPOUJOVB FO FM DPNQBDUP K &OUPODFT FYJTUFO a, b ∈ K UBMFT RVF f (a) = NÃY f Z f (b) = NÎO f  K

K

%FNPTUSBDJÓO 4FB s = TVQ f  7FBNPT RVF FYJTUF a ∈ K UBM RVF K

s = f (a) MP RVF EFNVFTUSB RVF s = NÃY f  1PS DBSBDUFSJ[BDJÓO EF K

TVQSFNP QBSB DBEB OBUVSBM n ≥ 1 FYJTUF xn ∈ K UBM RVF s − 1/n < f (xn ) ≤ s. $PNP TF SB[POÓ FO MB QSVFCB EFM UFPSFNB BOUFSJPS FYJTUFO VOB TVCTV DFTJÓO {xnk } Z a ∈ K UBMFT RVF f (xnk ) → f (a) -VFHP s − 1/nk < f (xnk ) ≤ s QBSB UPEP k ∈ N∗  5PNBOEP MÎNJUF DVBOEP k → ∞ FO MBT EFTJHVBMEBEFT QSFWJBT MMFHBNPT B MP EFTFBEP FT EFDJS s = f (a) 1BSB EFNPTUSBS MB FYJTUFODJB EF NÎO f TF BQMJDB MP RVF TF IB QSPCBEP B −f  FGFDUJWBNFOUF K

FYJTUF b ∈ K UBM RVF −f (b) = NÃY (−f ) = −NÎO f, K

K

EF EPOEF TF PCUJFOF MB DPODMVTJÓO 2VFEB BTÎ QSPCBEP FM UFPSFNB

 

&KFSDJDJP  4FB f : [a, b] −→ R DPOUJOVB Z UBM RVF f (x) > 0 QBSB UPEP x ∈ [a, b] 1SVFCF RVF FYJTUF VOB DPOTUBOUF m > 0 UBM RVF f (x) ≥ m QBSB UPEP x ∈ [a, b] &O MB QSVFCB EFM QSÓYJNP UFPSFNB TF FYIJCF VOB WF[ NÃT MB GVFS[B EFM BYJPNB EF DPNQMFUF[ FO MPT OÙNFSPT SFBMFT



-PT OÙNFSPT SFBMFT

5FPSFNB  4FBO a < b Z f VOB GVODJÓO DPO WBMPSFT FO R DPOUJOVB FO [a, b] Z UBM RVF f (a) < 0 < f (b) &OUPODFT FYJTUF c ∈ (a, b) UBM RVF f (c) = 0. %FNPTUSBDJÓO %FàOBNPT FM DPOKVOUP A = {t ∈ [a, b] : f (t) < 0} &TUF DPOKVOUP FT OP WBDÎP QVFT a ∈ A Z FT BDPUBEP TVQFSJPSNFOUF QPS b. -VFHP QPS FM BYJPNB EF DPNQMFUF[ FO MPT SFBMFT FYJTUF c ∈ R UBM RVF c = TVQ A 1SPCFNPT RVF c ∈ (a, b) Z RVF f (c) = 0 DPO MP DVBM TF UFSNJOB MB EFNPTUSBDJÓO EFM UFPSFNB 0CTFSWFNPT JOJDJBMNFOUF RVF f (c) ≤ 0 ZB RVF FYJTUF VOB TVDFTJÓO tn ∈ A UBM RVF tn → c Z DPNP f (tn ) < 0 QPS MB DPOUJOVJEBE EF f TF UJFOF RVF f (c) = MÎN f (tn ) ≤ 0 n→∞

%FCJEP B f (a) < 0 MB DPOUJOVJEBE EF f Z FM -FNB  MB GVODJÓO f DPOTFSWB FM TJHOP OFHBUJWP FO VOB WFDJOEBE EF a FT EFDJS FYJTUF δ1 > 0 UBM RVF f (t) < 0 QBSB UPEP t ∈ [a, a + δ1 ] /PUFNPT RVF {t ∈ [a, a + δ1 ] : f (t) < 0} ⊆ A Z QPS UBOUP TVQ{t ∈ [a, a + δ1 ] : f (t) < 0} ≤ c MP RVF JNQMJDB c ≥ a + δ1 > a %F NBOFSB TJNJMBS DPNP f (b) > 0 FYJTUF δ2 > 0 UBM RVF f (t) > 0 QBSB UPEP t ∈ [b − δ2 , b] "àSNBNPT RVF c < b − δ2 QVFT EF MP DPOUSBSJP b − δ2 ≤ c ≤ b Z BTÎ SFTVMUBSÎB MB DPOUSBEJDDJÓO f (c) > 0 %F FTUB BàSNBDJÓO TF TJHVF RVF c < b Z QPS DPOTJHVJFO UF a < c < b. 'JOBMNFOUF WFBNPT RVF f (c) = 0 CBTUB EFTDBSUBS RVF f (c) < 0 TJ PDVSSJFSB FTUP ÙMUJNP OVFWBNFOUF FM -FNB  JNQMJDBSÎB RVF FYJTUF δ3 > 0 TVàDJFOUFNFOUF QFRVFÒP UBM RVF f (t) < 0 QBSB UPEP t ∈ (c − δ3 , c + δ3 ) 'JKBSÎBNPT VO t ∈ (c, c + δ3 ) Z TFSÎB DMBSP RVF t ∈ [a, b] Z f (t ) < 0 3FTVMUBSÎB FOUPODFT RVF t ∈ A Z t ≤ c, VOB  DPOUSBEJDDJÓO &TUP QSVFCB RVF f (c) = 0  "QMJDBOEP FTUF UFPSFNB B MB GVODJÓO g(x) = f (x) − c TF QSVFCB FM TJHVJFOUF SFTVMUBEP $PSPMBSJP  5FPSFNB EFM WBMPS JOUFSNFEJP  4FBO a < b Z f VOB GVODJÓO DPO WBMPSFT FO R DPOUJOVB FO [a, b] Z UBM RVF f (a) < c < f (b) &OUPODFT FYJTUF x0 ∈ (a, b) UBM RVF f (x0 ) = c. &O FM DBTP f (a) > c > f (b) TF PCUJFOF MB NJTNB DPODMVTJÓO VTBOEP MB GVODJÓO −g(x) = c − f (x) &KFNQMP  4FB f : [0, 1] −→ [0, 1] VOB GVODJÓO DPOUJOVB &OUPODFT FYJTUF c ∈ [0, 1] UBM RVF f (c) = c. &TUF QVOUP c TF MMBNB QVOUP àKP EF MB GVODJÓO f  &GFDUJWBNFOUF TJ f (0) = 0 P f (1) = 1 MB DPODMVTJÓO FT JONFEJBUB 4VQPOHBNPT FOUPODFT RVF f (0) = 0 Z RVF f (1) = 1 $PNP f (0) ≥ 0 Z f (1) ≤ 1 FOUPODFT f (0) > 0 Z f (1) < 1 *OUSPEVDJNPT MB GVODJÓO BVYJMJBS g : [0, 1] −→ R EFàOJEB QPS g(x) = f (x) − x /PUFNPT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

RVF g(0) = f (0) > 0 Z g(1) = f (1) − 1 < 0. "QMJDBOEP FM UFPSFNB EFM WBMPS JOUFSNFEJP B MB GVODJÓO g FODPOUSBNPT c ∈ (0, 1) UBM RVF g(c) = 0 FT EFDJS f (c) = c. &KFSDJDJP  4FB f : [a, b] −→ [a, b] VOB GVODJÓO UBM RVF FYJTUF VOB DPOTUBOUF 0 ≤ c < 1 DPO MB TJHVJFOUF QSPQJFEBE QBSB UPEP x, y ∈ [a, b] TF DVNQMF RVF |f (x) − f (y)| ≤ c |x − y| 1SVFCF RVF f FT DPOUJOVB Z UJFOF B MP TVNP VO QVOUP àKP JBMBT`Qv2+iQ $POTVMUF TPCSF OÙNFSPT BMHFCSBJDPT Z OÙNFSPT USBTDFOEFOUFT EFàOJDJÓO FKFNQMPT QSFHVOUBT BCJFSUBT TPCSF FM QBSUJ DVMBS FUD



$BQÎUVMP  -PT OÙNFSPT DPNQMFKPT

%BEP RVF QBSB UPEP OÙNFSP SFBM x TF UJFOF RVF x2 ≥ 0 FOUPODFT MB 2 FDVBDJÓO x √ = −1 OP UJFOF TPMVDJÓO FO FM DBNQP R FO PUSBT QBMBCSBT OP FYJTUF −1 FO R 1PS FM &KFSDJDJP  EF MB TFDDJÓO  QÃH  FYJTUF VO DBNQP NÃT BNQMJP FO FM DVBM FTUB SBÎ[ DVBESBEB TÎ FYJTUF %JDIP DBNQP TFSÃ MMBNBEP FM EF MPT OÙNFSPT DPNQMFKPT

 $POTUSVDDJÓO EF MPT OÙNFSPT DPNQMFKPT %FàOJDJÓO  &M $POKVOUP C := R × R FT MMBNBEP FM DPOKVOUP EF MPT OÙNFSPT DPNQMFKPT &O FM DPOKVOUP EF MPT OÙNFSPT DPNQMFKPT EFàOJNPT WÎB FM FKFSDJDJP DJUBEP MBT TJHVJFOUFT PQFSBDJPOFT MBT DVBMFT EFOPUBNPT QPS DPNPEJEBE DPO MPT TÎNCPMPT DMÃTJDPT i + u Z i · u QBSB x1 , x2 , y1 , y2 ∈ R (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ), (x1 , y1 ) · (x2 , y2 ) = (x1 x2 − y1 y2 , x1 y2 + y1 x2 ). &OGBUJ[BNPT OVFWBNFOUF RVF QBSB OP IBDFS QFTBEB MB FTDSJUVSB IFNPT VTBEP FM NJTNP TÎNCPMP QBSB EFOPUBS MB TVNB Z FM QSPEVDUP FO DPOKVOUPT EJGFSFOUFT Z RVF FM DPOUFYUP QFSNJUF DMBSBNFOUF FMJNJOBS BNCJHÛFEBEFT (FPNÊUSJDBNFOUF VO OÙNFSP DPNQMFKP z = (x, y) TF SFQSFTFOUB DP NP FM FYUSFNP EFM WFDUPS FO FM QMBOP DBSUFTJBOP R×R DPO QVOUP JOJDJBM FO FM PSJHFO &M FKFSDJDJP RVF IFNPT WFOJEP NFODJPOBOEP SFTQBMEB FM TJHVJFOUF IFDIP 5FPSFNB  &M DPOKVOUP C EF MPT OÙNFSPT DPNQMFKPT EPUBEP EF MBT PQFSBDJPOFT QSFWJBNFOUF EFàOJEBT FT VO DBNQP



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

R

z R

'JHVSB  3FQSFTFOUBDJÓO HFPNÊUSJDB EF VO DPNQMFKP z = (x, y).

%BEP RVF (x, 0) + (y, 0) = (x + y, 0) Z RVF (x, 0) · (y, 0) = (xy, 0) QPEFNPT JEFOUJàDBS x DPO (x, 0) .ÃT QSFDJTBNFOUF TJ ˆ := {(x, 0) : x ∈ R} C ˆ QPS Φ(x) = (x, 0) FOUPODFT Φ FT VO JTPNPSàTNP Z TF EFàOF Φ : R → C &TUB BQMJDBDJÓO OPT QFSNJUF UBNCJÊO BàSNBS RVF C DPOUJFOF B R FO FM ˆ FM DVBM FT JTPNPSGP B R TFOUJEP EF RVF C DPOUJFOF VO TVCDPOKVOUP C $PO FTUP RVFEB EFNPTUSBEB MP TJHVJFOUF 1SPQPTJDJÓO  &M DPOKVOUP EF MPT OÙNFSPT SFBMFT FT VO TVCDPOKVOUP QSPQJP EFM DPOKVOUP EF MPT OÙNFSPT DPNQMFKPT 0CTFSWBDJÓO &T NVZ JNQPSUBOUF FM TJHVJFOUF DPNFOUBSJP TFB n ≥ 1 VO OÙNFSP OBUVSBM 4F EFàOF Rn DPNP FM DPOKVOUP EF UPEBT MBT nUVQMBT EF OÙNFSPT SFBMFT FT EFDJS Rn = {(x1 , x2 , . . . , xn ) : xi ∈ R, i = 1, 2, . . . , n}. 1BSB n = 1 UFOFNPT MPT OÙNFSPT SFBMFT Z QBSB n = 2 UFOFNPT MPT OÙNFSPT DPNQMFKPT 1BSB FTUPT ÙOJDPT WBMPSFT EF n TF EFNVFTUSB FO RVF Rn FT VO DBNQP %FàOJDJÓO  %BEP VO OÙNFSP DPNQMFKP z = (x, y) MMBNBNPT B x MB QBSUF SFBM EF z Z MB EFOPUBNPT QPS Re(z) -MBNBNPT B y MB QBSUF JNBHJOBSJB EF z MB DVBM EFOPUBNPT QPS Im(z). 6O OÙNFSP DPNQMFKP EF MB GPSNB (0, y) DPO y = 0 MP MMBNBNPT JNBHJOBSJP QVSP 1PEFNPT FOUPODFT BàSNBS RVF VO OÙNFSP DPNQMFKP FT VO OÙNFSP SFBM TJ TV QBSUF JNBHJOBSJB FT DFSP /PUFNPT UBNCJÊO RVF (x, y) = (x, 0) + (0, y) = (x, 0) + (0, 1) · (y, 0).



-PT OÙNFSPT DPNQMFKPT

%FàOJDJÓO  &M JNBHJOBSJP QVSP (0, 1) MP MMBNBNPT VOJEBE JNBHJOBSJB Z MP EFOPUBNPT QPS i FT EFDJS i := (0, 1). &TUB EFàOJDJÓO OPT QFSNJUF FTDSJCJS VO OÙNFSP DPNQMFKP z = (x, y) EF MB GPSNB z = x + iy 6TBNPT DVBMRVJFSB EF FTUBT EPT FTDSJUVSBT QBSB VO OÙNFSP DPNQMFKP 6O JNBHJOBSJP QVSP FT FOUPODFT EF MB GPSNB iy DPO y = 0 $VBOEP SFQSFTFOUBNPT HFPNÊUSJDBNFOUF VO OÙNFSP DPNQMFKP FTDSJ UP FO MB GPSNB z = x + iy FM NJTNP QMBOP DBSUFTJBOP EF EPT FKFT SFDUBO HVMBSFT SFDJCF PUSP OPNCSF EJBHSBNB EF "SHBOE EFOPNJOBEP B WFDFT QMBOP EF "SHBOE P QMBOP DPNQMFKP  "M FKF IPSJ[POUBM TF MF MMBNB FKF SFBM FO WF[ EF FKF EF MBT BCTDJTBT

Z BM WFSUJDBM TF MF MMBNB FKF JNBHJOBSJP FO WF[ EF FKF EF MBT PSEFOBEBT  6O DPNQMFKP z = x + iy HSÃàDBNFOUF MVDF BTÎ

Im(z)

y

x + iy x

Re(z)

'JHVSB  %JBHSBNB EF "SHBOE

0CTFSWFNPT RVF i2 = (0, 1) · (0, 1) = (−1, 0) = −1 Z UBNCJÊO (−i)2 = −1 %F FTUB NBOFSB −1 UJFOF EPT SBÎDFT DVBESBEBT FO C B TBCFS ±i. .ÃT BEFMBOUF QSFTFOUBSFNPT VO NÊUPEP QBSB IBMMBS SBÎDFT nÊTJNBT EF DVBMRVJFS OÙNFSP DPNQMFKP z. $PNP DPOTFDVFODJB EFM &KFSDJDJP  TF UJFOF FM TJHVJFOUF SFTVMUBEP FM DVBM NBOJàFTUB VOB EF MBT EJGFSFODJBT OPUBCMFT DPO MPT OÙNFSPT SFBMFT %JDIB EJGFSFODJB DPOTJTUF FO RVF OP IBZ PSEFO FO MPT OÙNFSPT DPNQMFKPT FT EFDJS OP IBZ EFTJHVBMEBEFT 1SPQPTJDJÓO  &M DBNQP C OP FT VO DBNQP PSEFOBEP %FàOJDJÓO  4FBO z1 = (x1 , y1 ) Z z2 = (x2 , y2 ) OÙNFSPT DPNQMFKPT %FDJNPT RVF z1 = z2 TJ x1 = x2 Z y1 = y2  &KFSDJDJP  4FBO z Z w DPNQMFKPT Z TVQPOHB RVF z + w Z zw TPO OÙNFSPT SFBMFT OFHBUJWPT 1SVFCF RVF z Z w EFCFO TFS OÙNFSPT SFBMFT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

%FàOJDJÓO  %BEP VO OÙNFSP DPNQMFKP z = x + iy EFàOJNPT TV DPNQMFKP DPOKVHBEP FM DVBM EFOPUBNPT QPS z DPNP z = x − iy /PUFNPT RVF VO DPNQMFKP Z TV DPOKVHBEP TPO TJNÊUSJDPT SFTQFDUP BM FKF SFBM

Im(z) w

z

Re(z) z

w 'JHVSB  $PNQMFKP Z TV DPOKVHBEP

"MHVOBT QSPQJFEBEFT EFM DPOKVHBEP TF QSFTFOUBO B DPOUJOVBDJÓO 1SPQPTJDJÓO  4FBO z Z w OÙNFSPT DPNQMFKPT &OUPODFT  z + w = z + w Z z · w = z · w.  4J w = 0 FOUPODFT z/w = z/w.  z = z ⇐⇒ z ∈ R. %FNPTUSBDJÓO 4FBO z = x+iy, w = u+iv. &M QSJNFS Z UFSDFS OVNFSBMFT TPO VOB DVFOUB EJSFDUB Z TJNQMF 1BSB FM TFHVOEP OVNFSBM VTBNPT FM BOUFSJPS QBSB PCUFOFS z = w · (z/w) = w · (z/w) EF MP DVBM TF TJHVF FM  SFTVMUBEP  %FàOJDJÓO  %BEP VO OÙNFSP DPNQMFKP z = x + iy EFàOJNPT TV NÓEVMP OPSNB MPOHJUVE P WBMPS BCTPMVUP FM DVBM + EFOPUBNPT QPS |z| DPNP FM OÙNFSP SFBM OP OFHBUJWP EBEP QPS |z| = x2 + y 2  &O FM DBTP FO FM RVF Im(z) = 0 FT EFDJS z = x ∈ R |z| DPJODJEF DPO FM WBMPS BCTPMVUP EF |x| EFàOJEP FO R. %F MB EFàOJDJÓO EF NÓEVMP EF VO OÙNFSP DPNQMFKP z = (x, y) TV SFQSFTFOUBDJÓO HFPNÊUSJDB FO FM QMBOP DBSUFTJBOP Z FM UFPSFNB EF



-PT OÙNFSPT DPNQMFKPT

1JUÃHPSBT WFNPT RVF |z| SFQSFTFOUB MB MPOHJUVE EF MB IJQPUFOVTB EFM USJÃOHVMP SFDUÃOHVMP DVZPT WÊSUJDFT TPO MPT QVOUPT (0, 0), (x, 0) Z (x, y). -B JEFB B DPOUJOVBDJÓO FT VTBS MBT OPDJPOFT FMFNFOUBMFT EF USJHPOP NFUSÎB QBSB QSFTFOUBS MB GPSNB USJHPOPNÊUSJDB EF VO OÙNFSP DPNQMFKP z = (x, y) = x + iy = 0. /PUFNPT RVF x = |z| DPT θ, y = |z| TJO θ, EPOEF θ ∈ (−π, π] FT FM ÙOJDP ÃOHVMP NFEJEP FO SBEJBOFT FOUSF FM FKF SFBM QPTJUJWP Z FM TFHNFOUP Oz. 0CTÊSWFTF RVF FYJTUFO JOàOJUPT ÃOHVMPT RVF TBUJTGBDFO MBT JHVBMEBEFT BOUFSJPSFT Z FT QPS FTP RVF BM ÙOJDP ÃOHVMP RVF WFSJàDB FTUBT JHVBMEBEFT FO FM JOUFSWBMP (−π, π] TF MF BTJHOB VO OPNCSF QBSUJDVMBS

y

(x, y) θ

x

x

θ (x, y) 'JHVSB  'PSNB USJHPOPNÊUSJDB

%FàOJDJÓO  4FB z = (x, y) VO OÙNFSP DPNQMFKP OP OVMP &M ÙOJ DP ÃOHVMP θ ∈ (−π, π] NFEJEP FO SBEJBOFT UBM RVF x = |z| DPT θ y = |z| TJO θ, TF MMBNB BSHVNFOUP QSJODJQBM EF z Z MP EFOPUBNPT QPS "SH z -PT EFNÃT ÃOHVMPT RVF TBUJTGBDFO EJDIBT JHVBMEBEFT TF MMBNBO BSHVNFOUPT Z EFOPUBNPT QPS BSH z FM DPOKVOUP EF UPEPT MPT BSHVNFOUPT -B FYQSFTJÓO z = |z| DPT θ + i|z| TJO θ TF MMBNB GPSNB USJHPOPNÊUSJDB P QPMBS EFM OÙNFSP DPNQMFKP z. 1VFTUP RVF x2 ≤ x2 + y 2 = |z|2 FOUPODFT |Re(z)| ≤ |z|. %F NBOFSB TJNJMBS |Im(z)| ≤ |z|. 5BNCJÊO TF UJFOF RVF Re(z) =

z+z , 2

Im(z) =

z−z · 2i



&T JONFEJBUP EF MBT EFàOJDJPOFT Z MBT NBOJQVMBDJPOFT BMHFCSBJDBT EJT QPOJCMFT FO MPT DBNQPT RVF |z|2 = zz. &TUP OPT QFSNJUF IBMMBS VOB FY QSFTJÓO IBTUB BIPSB OP FTUBCMFDJEB QBSB FM JOWFSTP EF z = (x, y) = 0 &O FGFDUP DPNP 1 z = z −1 = 2 , z |z| 

5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

−y  x . , 2 2 + y x + y2 6TBOEP MB EFàOJDJÓO EF JHVBMEBE EF EPT DPNQMFKPT TF QVFEF FTUBCMF DFS EF PUSB NBOFSB MB FYQSFTJÓO QBSB FM SFDÎQSPDP EF z = (x, y) = 0 TFB z −1 = (u, v) EJDIP JOWFSTP FOUPODFT (x, y) · (u, v) = (1, 0) EF EPOEF TF PCUJFOF MB FYQSFTJÓO EBEB BSSJCB QBSB z −1 . 6OB EF MBT WFOUBKBT EFM DPODFQUP EF DPOKVHBEP EF VO DPNQMFKP FT RVF QFSNJUF DPOPDFS FO MB GPSNB FTUÃOEBS FM DPDJFOUF EF EPT DPNQMFKPT

FOUPODFT z −1 =



x2

5−i . 3 + 2i &M QVOUP DMBWF FT VTBS FM DPOKVHBEP EFM EFOPNJOBEPS EF MB TJHVJFOUF NBOFSB &KFNQMP  )BMMBS FO MB GPSNB FTUÃOEBS a + ib FM DPDJFOUF

5−i 5 − i 3 − 2i 1 = = (13 − 13i) = 1 − i. 3 + 2i 3 + 2i 3 − 2i 13 &KFSDJDJP  &TDSJCB FM SFTVMUBEP FO MB GPSNB a + ib : 2+i 4+i − · 3 − i 1 + 2i   3 + 2i 5 − 2i C

+ · 1+i i−1

B

&KFSDJDJP  &ODVFOUSF UPEPT MPT WBMPSFT QPTJCMFT EF in WBSJBOEP n ∈ N. &KFSDJDJP  %FNVFTUSF MB JEFOUJEBE EFM QBSBMFMPHSBNP QBSB OÙNFSPT DPNQMFKPT |z + w|2 + |z − w|2 = 2(|z|2 + |w|2 ). &KFSDJDJP  4FBO z Z w OÙNFSPT DPNQMFKPT √ UBMFT RVF |z| ≤ 1, |w| ≤ 1 Z |z − w| ≥ 1. %FNVFTUSF RVF |z + w| ≤ 3. √ &KFSDJDJP  4FB z = x + iy ∈ C. 1SVFCF RVF |x| + |y| ≤ 2 |z|.    z−a   < 1. &KFSDJDJP  4J |z| < 1 Z |a| < 1 QSVFCF RVF  1 − az &KFSDJDJP  &TDSJCB FO UÊSNJOPT EF MB WBSJBCMF z MB FDVBDJÓO EF MB IJQÊSCPMB x2 − y 2 = 1. -B NJTNB QSVFCB EF $ FO FM 5FPSFNB  EFNVFTUSB 1SPQPTJDJÓO  4FBO z1 Z z2 OÙNFSPT DPNQMFKPT UBMFT RVF z1 z2 = 0. &OUPODFT z1 = 0 Ó z2 = 0.



-PT OÙNFSPT DPNQMFKPT

&M NÓEVMP EF VO OÙNFSP DPNQMFKP TBUJTGBDF MBT TJHVJFOUFT QSPQJFEBEFT 5FPSFNB  4FBO z Z w OÙNFSPT DPNQMFKPT &OUPODFT B |zw| = |z| |w|. 4J w = 0 FOUPODFT |z/w| = |z|/|w|. C |z + w| ≤ |z| + |w| %FTJHVBMEBE USJBOHVMBS    D |z| − |w| ≤ |z − w| $POUJOVJEBE EFM NÓEVMP  %FNPTUSBDJÓO B /PUFNPT RVF |zw|2 = (zw)zw = zzww = |z|2 |w|2 JNQMJDB MB BàSNBDJÓO -B TFHVOEB QBSUF FT DPOTFDVFODJB EF MB QSJ NFSB QVFT |z| = |w(z/w)| = |w| |z/w|. C $PNP FO FM OVNFSBM BOUFSJPS MB DMBWF FT QSPCBS RVF MPT DVBESBEPT TBUJTGBDFO VOB EFTJHVBMEBE EF MB DVBM TF EFTQSFOEB FM SFTVMUBEP EFTFBEP |z + w|2 = (z + w)(z + w) = |z|2 + zw + wz + |w|2 = |z|2 + zw + zw + |w|2 = |z|2 + 2Re(zw) + |w|2 2

≤ |z| + 2|zw| + |w|

QPS 

2

= |z|2 + 2|z| |w| + |w|2

QPS (a)

2

= (|z| + |w|) . D #BTUB BQMJDBS MB EFTJHVBMEBE USJBOHVMBS BTÎ |z| = |z − w + w| ≤ |z − w| + |w| ⇒ |z| − |w| ≤ |z − w|. $BNCJBOEP MPT SPMFT EF z Z w TF PCUJFOF |w| − |z| ≤ |z − w| Z BM DPNCJOBS FTUBT EPT EFTJHVBMEBEFT TF PCUJFOF FM SFTVMUBEP 2VFEB BTÎ QSPCBEP FM UFPSFNB

 

-B JHVBMEBE FO MB EFTJHVBMEBE USJBOHVMBS TF UJFOF DVBOEP MPT DPN QMFKPT JOWPMVDSBEPT TPO QBSBMFMPT Z FO MB NJTNB EJSFDDJÓO : SFDÎQSPDB NFOUF &KFNQMP  4FBO z Z w DPNQMFKPT OP OVMPT 1SPCFNPT RVF |w + z| = |w| + |z|



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

TJ Z TPMP TJ FYJTUF t > 0 UBM RVF w = tz. -B DPOEJDJÓO TVàDJFOUF FT JONFEJBUB QVFT |w + z| = |tz + z| = (t + 1)|z| = |tz| + |z| = |w| + |z|. 4VQPOHBNPT BIPSB RVF |w + z| = |w| + |z| &MFWBOEP BM DVBESBEP Z SFBMJ[BOEP MPT DÃMDVMPT MMFHBNPT B MB JHVBMEBE Re(z w) = |zw| ≥ 0.



&TDSJCBNPT z = x + iy Z w = u + iv 1PS MB JHVBMEBE Z MB EFTJHVBMEBE FO  TF UJFOF RVF Im(zw) = yu−vx = 0 Z ux+vy ≥ 0 SFTQFDUJWBNFOUF 7BNPT B DPOTJEFSBS EPT DBTPT v = 0 Z v = 0. %FM QSJNFS DBTP MB IJQÓUFTJT Z MB JHVBMEBE QSFWJB TF TJHVF RVF y = 0 Z QPS UBOUP z = x Z w = u. $PNP u = 0 FOUPODFT IBDJFOEP t = x/u UFOFNPT RVF z = x = (x/u)u = tw. 0CTFSWFNPT RVF FO FTUF DBTP Re(zw) = ux > 0 Z BTÎ t > 0, RVFEBOEP QSPCBEB MB DPOEJDJÓO OFDFTBSJB FO FTUF DBTP 4J v = 0 FOUPODFT x = (u/v)y Z EFàOJFOEP t = y/v TF PCUJFOF RVF z = x+iy = (y/v)(u+iv) = tw TPMP SFTUB WFS RVF t > 0 EBEP RVF 0 ≤ Re(zw) = ux + vy Z v 2 > 0 FOUPODFT (ux + vy)/v 2 ≥ 0 FT EFDJS (u2 /v 2 )t + t ≥ 0 MP DVBM FRVJWBMF B EFDJS RVF t|w|2 ≥ 0. 4J t = 0 FOUPODFT y = x = z = 0 MP DVBM DPOUSBEJDF MB IJQÓUFTJT 1PS UBOUP t > 0.

 'ÓSNVMB EF %F .PJWSF -B GPSNB USJHPOPNÊUSJDB EF VO OÙNFSP DPNQMFKP QFSNJUF FO PDBTJPOFT SFBMJ[BS DJFSUBT PQFSBDJPOFT DPNP MB QPUFODJBDJÓO Z MB SBEJDBDJÓO EF VOB NBOFSB NÃT TJNQMF *OJDJBNPT FTUB TFDDJÓO EFTDSJCJFOEP MB FYQSFTJÓO USJHPOPNÊUSJDB DPSSFTQPOEJFOUF BM QSPEVDUP Z BM DPDJFOUF EF EPT DPN QMFKPT OP OVMPT Z MVFHP MB QPUFODJBDJÓO QBSB B QBSUJS EF BIÎ EFEVDJS FM UFPSFNB EF %F .PJWSF 'JOBMNFOUF OPT PDVQBNPT EF MB SBEJDBDJÓO 4FBO z1 = |z1 |(DPT θ1 + i TFOθ1 ) Z z2 = |z2 |(DPT θ2 + i TFOθ2 ) EPT OÙNFSPT DPNQMFKPT OP OVMPT &OUPODFT z1 · z2 = |z1 z2 |(DPT θ1 DPT θ2 − TFOθ1 TFOθ2 + i(DPT θ1 TFOθ2 + TFOθ1 DPT θ2 ))   = |z1 z2 | DPT(θ1 + θ2 ) + i TFO(θ1 + θ2 ) .



-B FYQSFTJÓO  MB QPEFNPT JOUFSQSFUBS EJDJFOEP EF NBOFSB MJHFSB Z OP QSFDJTB RVF QBSB IBMMBS FM QSPEVDUP EF EPT DPNQMFKPT FTDSJUPT FO GPSNB USJHPOPNÊUSJDB CBTUB NVMUJQMJDBS TVT NÓEVMPT Z TVNBS TVT BSHVNFOUPT



-PT OÙNFSPT DPNQMFKPT

&KFNQMP  4FB z = |z|(DPT θ + i TFOθ) = 0 )BMMFNPT FM QSPEVDUP iz OPUFNPT QSJNFSP RVF i = cos(π/2) + i TFO(π/2) Z QPS 

TF PCUJFOF RVF iz = |z|(DPT(θ + π/2) + i TFO(θ + π/2) = |z|(−TFOθ + i DPT θ). 

&O SFBMJEBE FM PCKFUJWP QSJODJQBM EF FTUF FKFNQMP FT JMVTUSBS RVF FM QSP EVDUP EF VO DPNQMFKP z = 0 QPS MB VOJEBE JNBHJOBSJB i FT PUSP DPNQMFKP FM DVBM TF PCUJFOF SPUBOEP FM i WFDUPS u BTPDJBEP B z FO VO ÃOHVMP EF π/2 FO TFOUJEP BOUJIPSBSJP &TUP TF EFEVDF EF MB QSJNFSB JHVBMEBE FO  

Im(z) z iz Re(z)

'JHVSB  &GFDUP HFPNÊUSJDP iz

7FBNPT BIPSB FM DPDJFOUF QBSB DPNQMFKPT z1 Z z2 OP OVMPT UFOJFOEP FO DVFOUB RVF FM DPOKVHBEP EF z2 FTUÃ EBEP QPS z2 = |z2 |(DPT θ2 − i TFOθ2 ), UFOFNPT RVF z1 z1 z2 = z2 |z2 |2 |z1 | (DPT θ1 DPT θ2 + TFOθ1 TFOθ2 + i(DPT θ2 TFOθ1 − TFOθ2 DPT θ1 )) = |z2 |  |z1 |  = DPT(θ1 − θ2 ) + i TFO(θ1 − θ2 ) . 

|z2 | -B FYQSFTJÓO  EJDF EF NBOFSB MJHFSB Z OP QSFDJTB RVF QBSB IBMMBS FM DPDJFOUF EF EPT DPNQMFKPT FTDSJUPT FO GPSNB USJHPOPNÊUSJDB CBTUB FTUBCMFDFS FM DPDJFOUF EF TVT NÓEVMPT Z SFTUBS TVT BSHVNFOUPT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFSDJDJP  1SVFCF VTBOEP JOEVDDJÓO RVF TJ z = |z|(DPT θ + i TFOθ) = 0 FOUPODFT QBSB UPEP n ∈ N TF WFSJàDB MB JHVBMEBE   z n = |z|n DPT(nθ) + i TFO(nθ) . %F FTUF FKFSDJDJP TF PCUJFOF MB WBMJEF[ QBSB FOUFSPT OFHBUJWPT FO FGFDUP TFBO n ∈ Z DPO n ≤ −1 Z m = −n. &OUPODFT m FT VO OBUVSBM DPO m ≥ 1. -VFHP TJ z = |z|(DPT θ + i TFOθ) = 0 BM BQMJDBS FM FKFSDJDJP QSFWJP Z FM IFDIP EF RVF z −1 = z/|z|2 UFOFNPT RVF   −1 z n = z −m = (z m )−1 = |z|m DPT(mθ) + i TFO(mθ)  −1 = |z|−m DPT(mθ) + i TFO(mθ)   = |z|n DPT(mθ) − i TFO(mθ)   = |z|n DPT(−mθ) + i TFO(−mθ)   = |z|n DPT(nθ) + i TFO(nθ) . &O QBSUJDVMBS TJ |z| = 1 DPODMVJNPT FM TJHVJFOUF JNQPSUBOUF SFTVMUBEP 5FPSFNB  'ÓSNVMB EF %F .PJWSF  1BSB UPEP SFBM θ Z QBSB UPEP FOUFSP n TF UJFOF RVF (DPT θ + i TFOθ)n = DPT(nθ) + i TFO(nθ). 6OB BQMJDBDJÓO JONFEJBUB EF MB GÓSNVMB EF %F .PJWSF FT MB EFEVDDJÓO EF VOB SFHMB QBSB IBMMBS MB SBÎ[ nÊTJNB n ≥ 2 EF VO OÙNFSP DPNQMFKP 5FPSFNB  4FB z = 0 VO OÙNFSP DPNQMFKP DPO "SH z = θ. &OUPODFT z QPTFF FYBDUBNFOUF n SBÎDFT nÊTJNBT EJGFSFOUFT MBT DVBMFT FTUÃO EBEBT QPS  θ + 2kπ  θ + 2kπ )+i TFO( ) , DPO k = 0, 1, . . . , n−1. z 1/n = |z|1/n DPT( n n %FNPTUSBDJÓO 4FBO z = |z|(DPT θ + i TFOθ) Z w = |w|(DPT α + i TFOα) UBMFT RVF z 1/n = w FTUP FT wn = z. -VFHP QPS MB GÓSNVMB EF %F .PJWSF UFOFNPT RVF   

|z|(DPT θ + i TFOθ) = |w|n DPT(nα) + i TFO(nα) .





&O IPOPS B "CSBIBN %F .PJWSF 

B RVJFO TF BUSJCVZF TV EFTDVCSJNJFOUP &SB VO GSBODÊT SFGVHJBEP FO -POESFT RVF DVSTÓ TVT FTUVEJPT DPO NVDIBT EJàDVMUBEFT *OUSPEVKP DBOUJEBEFT JNBHJOBSJBT FO USJHPOPNFUSÎB

-PT OÙNFSPT DPNQMFKPT

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5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

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/PUFNPT RVF EPT EF FMMBT EJàFSFO ZB TFB QPS VOB MFUSB P QPS VOB PS EFOBDJÓO EJGFSFOUF EF MBT MFUSBT -BT WBSJBDJPOFT EF USFT FO USFT TPO FO UPUBM 24 MP DVBM TF FYQMJDB BTÎ MB QSJNFSB MFUSB TF PCUJFOF EF  NBOFSBT VOB WF[ TF UFOHB FTUB RVFEBO  QPTJCJMJEBEFT QBSB MB TFHVOEB MFUSB Z àOBMNFOUF QBSB MB QPTJDJÓO UFSDFSB RVFEBO EPT PQDJPOFT 1PS FM QSJODJ QJP GVOEBNFOUBM EFM DPOUFP UFOFNPT 4 · 3 · 2 = 24 WBSJBDJPOFT UPNBEBT EF B USFT 6OB GÓSNVMB QBSB DBMDVMBS MBT WBSJBDJPOFT EF n PCKFUPT UPNBEPT EF r FO r TF EFEVDF EF MB TJHVJFOUF NBOFSB OPUFNPT JOJDJBMNFOUF RVF FTUP FRVJWBMF B FODPOUSBS FM OÙNFSP EF PSEFOBNJFOUPT EF r PCKFUPT DVBOEP EJTQPOFNPT EF n EF FMMPT &M QSJNFS MVHBS TF QVFEF FTDPHFS EF n QPTJCJ MJEBEFT Z DVBOEP IB TJEP FTDPHJEP EF BMHVOB EF FMMBT FM TFHVOEP MVHBS



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

QVFEF TFS MMFOBEP EF BMHVOB EF MBT n − 1 PQDJPOFT EJTQPOJCMFT $PNP DBEB NBOFSB EF PDVQBS FM QSJNFS MVHBS QVFEF BTPDJBSTF DPO DBEB NB OFSB EF PDVQBS FM TFHVOEP QPS FM QSJODJQJP GVOEBNFOUBM EFM DPOUFP FM OÙNFSP EF NBOFSBT FO RVF TF QVFEFO PDVQBS MPT EPT QSJNFSPT MVHBSFT FTUÃ EBEP QPS n(n − 1). "IPSB CJFO DVBOEP MPT EPT QSJNFSPT MVHBSFT IBO TJEP PDVQBEPT EF DVBMRVJFSB EF FTUBT NBOFSBT FM UFSDFS MVHBS QVFEF MMFOBSTF EF n − 2 GPSNBT 3B[POBOEP DPNP BOUFT FM OÙNFSP EF NBOFSBT FO RVF QVFEFO TFS PDVQBEPT USFT MVHBSFT FT n(n−1)(n−2). 1SPDFEJFOEP EF FTUB GPSNB Z OPUBOEP RVF TF JOUSPEVDF VO OVFWP GBDUPS QPS DBEB MVHBS PDVQBEP UFOFNPT RVF FM OÙNFSP EF NBOFSBT EF PDVQBS MPT r QVFTUPT FT FM QSPEVDUP EF MPT r GBDUPSFT n(n − 1)(n − 2) · · · (n − (r − 1)). 4J EFOPUBNPT QPS n Vr FM OÙNFSP EF MBT WBSJBDJPOFT EF n PCKFUPT UPNBEPT EF r FO r UFOFNPT RVF n

Vr = n(n − 1)(n − 2) · · · (n − r + 1).

&M TVQFSÎOEJDF n EFOPUB FM OÙNFSP EF PCKFUPT EJTQPOJCMFT Z FM TVCÎOEJDF r JOEJDB UBOUP FM OÙNFSP EF FMFNFOUPT FO DBEB VOB EF MBT BHSVQBDJPOFT FO FM PSEFOBNJFOUP DPNP FM OÙNFSP EF GBDUPSFT FO FM QSPEVDUP 0CTFSWFNPT RVF DVBOEP r = n FT EFDJS FO FM DBTP EF QFSNVUBDJPOFT EF n PCKFUPT MBT DVBMFT EFOPUBNPT QPS Pn FM OÙNFSP EF FTUBT FTUÃ EBEP QPS n(n − 1)(n − 2) · · · 1 = n! $PODSFUBOEP Pn = n! &KFNQMP "  y$VÃOUPT OÙNFSPT EJGFSFOUFT EF  DJGSBT QVFEFO PCUFOFSTF DPO MPT OVFWF EÎHJUPT 1, 2, 3, . . . , 9 $PNP MB JEFB FT PDVQBS TJFUF MVHBSFT JNQPSUB FM PSEFO EJTQPOJFOEP EF  PCKFUPT FTUBNPT IBCMBOEP EF WBSJBDJPOFT EF  FMFNFOUPT UPNBEPT BM NJTNP UJFNQP EF  FO  FT EFDJS 9 V7 = 9 · 8 · 7 · 6 · 5 · 4 · 3 = 181440.  y$VÃOUBT PSEFOBDJPOFT QVFEFO UFOFSTF DPO MB MFUSBT EF MB QBMBCSB +m/2`MQ y$VÃOUBT DPNJFO[BO DPO + Z UFSNJOBO DPO Q -B QSJ NFSB QBSUF TF USBUB EF QFSNVUBS MBT PDIP MFUSBT EF MB QBMBCSB DVBEFSOP Z QPS UBOUP MB SFTQVFTUB FT 8! = 40320. 1BSB MB TF HVOEB QBSUF WBO B RVFEBS àKBT MB QSJNFSB Z ÙMUJNB MFUSBT Z FO DPOTFDVFODJB CBTUB QFSNVUBS MBT PUSBT TFJT MFUSBT MP RVF OPT EB VO OÙNFSP EF 1 · 6! · 1 = 720 FT EFDJS 720 QFSNVUBDJPOFT EF MBT MFUSBT EF MB QBMBCSB DVBEFSOP DPNJFO[BO DPO + Z UFSNJOBO DPO Q



"QÊOEJDF

 y%F DVÃOUBT NBOFSBT QVFEFO PSEFOBSTF  MJCSPT FO VO FTUBOUF TJ USFT MJCSPT EFUFSNJOBEPT TJFNQSF EFCFO FTUBS KVOUPT 7BNPT B DPOTJEF SBS MPT USFT MJCSPT RVF TJFNQSF EFCFO FTUBS KVOUPT DPNP VO TPMP UPNP Z BTÎ DPOWFSUJNPT FM QSPCMFNB FO VOB QFSNVUBDJÓO EF  PCKFUPT MP DVBM EB 4! NBOFSBT EF PSEFOBSMPT 1FSP QPS DBEB VOB EF FTUBT EFCFNPT UFOFS FO DVFOUB RVF FM UPNP EF MPT USFT MJCSPT OPT PGSFDF 3! PSEFOBDJPOFT &O DPOTFDVFODJB FYJTUFO 4! · 3! = 144 NBOFSBT EF PSEFOBS MPT  MJCSPT EFKBOEP USFT MJCSPT EFUFSNJOBEPT TJFNQSF KVOUPT &KFSDJDJP "  5SFT NBUSJNPOJPT TF SFÙOFO QBSB DFMFCSBS FM BOJWFSTBSJP EF VOP EF FMMPT %FTFBO RVF MFT UPNFO VOB GPUPHSBGÎB EF GPSNB RVF FTUÊO UPEPT MPT IPNCSFT KVOUPT Z UBNCJÊO MBT NVKFSFT y%F DVÃOUBT GPSNBT EJTUJOUBT QVFEFO IBDFSMP  $PO MPT EÎHJUPT QBSFT yDVÃOUPT OÙNFSPT NFOPSFT RVF  TF QVFEFO GPSNBS  &O MBT WBSJBDJPOFT TJO SFQFUJDJÓO RVF QPEFNPT GPSNBS DPO MBT OVFWF DJGSBT TJHOJàDBUJWBT FT EFDJS DPO MBT DJGSBT 1, 2, . . . , 9 UPNBEBT EF USFT FO USFT yDVÃOUBT WFDFT FTUÃ MB DJGSB   4J TF TVQPOFO PSEFOBEBT UPEBT MBT QFSNVUBDJPOFT RVF TF QVFEFO GPS NBS DPO MBT DJGSBT 1, 2, 3, 5, 8, 9 FO PSEFO DSFDJFOUF yRVÊ MVHBS PDV QB MB QFSNVUBDJÓO  "IPSB OPT PDVQBNPT EF QSFTFOUBS MBT BHSVQBDJPOFT FO MBT DVBMFT FM PSEFO OP JNQPSUB %FàOJDJÓO " $BEB VOP EF MPT HSVQPT RVF QVFEFO GPSNBSTF UPNBOEP BMHVOPT P UPEPT MPT PCKFUPT EJTQPOJCMFT EF NPEP RVF EPT DVBMFTRVJFSB EF FMMPT EJàFSFO FO BMHÙO PCKFUP TF MMBNB VOB DPNCJOBDJÓO $PNP FKFNQMP OPUFNPT RVF MBT DPNCJOBDJPOFT RVF QVFEFO GPSNBS TF DPO MBT MFUSBT a, b, c, d UPNBEBT EF EPT FO EPT TPO MBT TJHVJFOUFT ab, ac, ad, bc, bd, cd EPOEF EPT DVBMFTRVJFSB EF FMMBT EJàFSFO FO VOB MF USB 3FDBMDBNPT RVF QBSB GPSNBS DPNCJOBDJPOFT TPMP JOUFSFTB FM OÙNFSP EF FMFNFOUPT RVF DPOUJFOF DBEB TFMFDDJÓO NJFOUSBT RVF QBSB GPSNBS WB SJBDJPOFT UFOFNPT RVF DPOTJEFSBS UBNCJÊO FM PSEFO EF MPT FMFNFOUPT RVF GPSNBO DBEB PSEFOBDJÓO "TÎ ab Z ba TPO MB NJTNB DPNCJOBDJÓO QFSP TPO WBSJBDJPOFT EJTUJOUBT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP " %F MBT MFUSBT a, b, c, d UPNBEBT EF B USFT PCUFOFNPT MBT DPNCJOBDJPOFT abc, abd, bcd Z acd $BEB VOB EF FTUBT DPNCJOBDJPOFT BENJUF 3! PSEFOBDJPOFT Z QPS UBOUP FM UPUBM EF WBSJBDJPOFT EF MBT DVBUSP MFUSBT UPNBEBT EF B USFT FO USFT TPO  DPNP ZB TBCÎBNPT 7BNPT B EFEVDJS BIPSB VOB GÓSNVMB RVF OPT QFSNJUB IBMMBS FM OÙNFSP EF DPNCJOBDJPOFT EF n PCKFUPT EJGFSFOUFT UPNBEPT EF r FO r %FOPUFNPT UBM OÙNFSP QPS n Cr  $PO n = 4 Z r = 3 FM FKFNQMP BOUFSJPS TVHJFSF RVF n Cr · r! = n Vr . "àSNBNPT RVF FO SFBMJEBE FTUB JHVBMEBE FT DJFSUB FO FGFDUP DPNP DBEB VOB EF MBT DPNCJOBDJPOFT DPOUJFOF r FMFNFOUPT EJGFSFOUFT RVF DPNP TBCFNPT QVFEFO TFS PSEFOBEPT FOUSF TÎ EF r! NB OFSBT FOUPODFT FM QSPEVDUP n Cr · r! FT JHVBM BM OÙNFSP EF WBSJBDJPOFT EF n PCKFUPT UPNBEPT EF B r FM DVBM IFNPT EFOPUBEP QPS n Vr  %F FTUB NBOFSB RVFEB QSPCBEB MB BàSNBDJÓO 4FHVJEBNFOUF WBNPT B PCUFOFS MB GÓSNVMB QSPNFUJEB Z QBSB FMMP SFDPSEFNPT RVF n

Vr = n(n − 1)(n − 2) · · · (n − r + 1) n(n − 1)(n − 2) · · · (n − r + 1)(n − r)! = (n − r)! n! = , (n − r)!

EF MP DVBM TF TJHVF RVF n

Cr =

n! · r!(n − r)!

"

4J FO MB GÓSNVMB " IBDFNPT n = r Z DPNP n Cn = 1 FOUPODFT MB EFàOJDJÓO 0! = 1 RVFEB QMFOBNFOUF NPUJWBEB 0CTFSWFNPT UBNCJÊO RVF FO EJDIB GÓSNVMB BM DBNCJBS r QPS n − r PCUFOFNPT RVF n Cr = n Cn−r . &KFNQMP " 6O FTUVEJBOUF UJFOF RVF DPOUFTUBS  EF MBT  QSFHVO UBT EF VO FYBNFO y%F DVÃOUBT GPSNBT EJGFSFOUFT QVFEF DPOUFTUBS y: TJ MBT USFT QSJNFSBT TPO PCMJHBUPSJBT y: TJ EF MBT DJODP QSJNFSBT IB EF DPOUFTUBS B DVBUSP 1BSB MB QSJNFSB QBSUF OP JNQPSUB FM PSEFO TF USBUB EF IBMMBS UPEPT MPT HSVQPT QPTJCMFT EF  FMFNFOUPT EF VO UPUBM EF  10! FT EFDJS IBZ 10 C8 = 2!8! = 45 QPTJCMFT HSVQPT EJGFSFOUFT EF  QSFHVO UBT 1BSB MB TFHVOEB QBSUF DPNP PCMJHBUPSJBNFOUF EFCF SFTQPOEFS MBT USFT QSJNFSBT FOUPODFT EF MBT TJFUF SFTUBOUFT GPSNB HSVQPT EF B DJODP 7! = 21 NBOFSBT Z BTÎ QVFEF SFTQPOEFS MBT  QSFHVOUBT EF 7 C5 = 5!2! 'JOBMNFOUF QBSB SFTPMWFS FTUB JORVJFUVE OPUFNPT RVF IBZ EPT FWFOUPT JOWPMVDSBEPT FM QSJNFSP DPOTJTUF FO TFMFDDJPOBS HSVQPT EF B  EF DJODP EJTQPOJCMFT Z FM TFHVOEP FWFOUP DPOTJTUF FO FTDPHFS HSVQPT EF B  EF



"QÊOEJDF

MBT DJODP SFTUBOUFT &M QSJNFS FWFOUP QVFEF EBSTF EF 5 C4 = 5 NBOFSBT Z FM TFHVOEP UBNCJÊO EF 5 C4 NBOFSBT 1PS FM QSJODJQJP GVOEBNFOUBM EFM DPOUFP IBZ  NBOFSBT EF SFTQPOEFS FM FYBNFO CBKP FTUBT DPOEJDJPOFT &KFNQMP " $BMDVMFNPT FM OÙNFSP EF EJBHPOBMFT RVF UJFOF VO QP MÎHPOP EF  MBEPT Z FO HFOFSBM QBSB VOP EF O MBEPT -BT EJBHPOBMFT TF PCUJFOFO VOJFOEP WÊSUJDFT QFSP EFCFNPT SFTUBS MPT  MBEPT EFM QP MÎHPOP ZB RVF FTUPT OP TPO EJBHPOBMFT $PO MPT  WÊSUJDFT TF GPSNBO 12 C = 66 TFHNFOUPT Z BTÎ FM OÙNFSP EF EJBHPOBMFT EF VO EPEFDÃHPOP 2 FT 66 − 12 = 54 1BSB FM QPMÎHPOP EF n MBEPT BQMJDBNPT FM NJTNP SB[P OBNJFOUP FM OÙNFSP EF TFHNFOUPT RVF TF DPOTJHVFO DPO MPT n WÊSUJDFT FTUÃ EBEP QPS n C2 Z FO DPOTFDVFODJB FM OÙNFSP EF EJBHPOBMFT FT n

C2 − n =

n(n − 1) n(n − 3) n! −n= −n= · 2(n − 2)! 2 2

&KFNQMP " &O VO UFTU EF  QSFHVOUBT DPO EPT PQDJPOFT yEF DVÃO UBT GPSNBT QVFEFO NBSDBSTF MBT QSFHVOUBT QBSB RVF BM NFOPT  FTUÊO DPSSFDUBT &M QSPCMFNB FT FRVJWBMFOUF B FODPOUSBS FM OÙNFSP EF NBOFSBT EF SFTQPOEFS FM FYBNFO GBMMBOEP FO B MP TVNP USFT QSFHVOUBT FTUP FT    Ó  JODPSSFDUBT &TUF OÙNFSP FTUÃ EBEP QPS 20

C3 +

20

C2 +

20

C1 +

20

C0 = 1351.

&KFSDJDJP " %F  JOHMFTFT Z  BNFSJDBOPT TF WB B GPSNBS VO DPNJUÊ EF  JOUFHSBOUFT y%F DVÃOUBT NBOFSBT QVFEF IBDFSTF TJ DPNP NÎOJNP EFCFO FTUBS  BNFSJDBOPT RVF TF BDPTUVNCSB UBNCJÊO VTBS MB OPUBDJÓO n 7BMF MB QFOB NFODJPOBS n r QBSB EFOPUBS Cr . &KFSDJDJP "  4FBO n, r ∈ N DPO r + 1 ≤ n %FNVFTUSF RVF

n+1 r+1

=



n r+1



+

n  r

.

 1SVFCF RVF TJ r ∈ N FTUÃ àKP FOUPODFT  n   r+k k=0

k

 =

 r+n+1 . n

 %FNVFTUSF RVF QBSB DBEB n ∈ N TF DVNQMF RVF

  n  n k = n2n−1  k k=0



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

" $POKVOUPT àOJUPT F JOàOJUPT %FàOJDJÓO " J 4FBO A Z B EPT DPOKVOUPT OP WBDÎPT %FDJNPT RVF A Z B TPO FRVJQPUFOUFT P RVF UJFOFO MB NJTNB QPUFODJB TJ FYJTUF VOB GVODJÓO CJZFDUJWB f EF A FO B 4J A Z B TPO FRVJQPUFOUFT FTDSJCJNPT A ∼ B. JJ %FàOJNPT FM TFHNFOUP JOJDJBM EF OÙNFSPT OBUVSBMFT DPNP FM DPO KVOUP Jn := {k ∈ N : 1 ≤ k ≤ n} QBSB BMHÙO n ∈ N. /ÓUFTF RVF Jn = Jm TJ Z TPMP TJ Jn ⊆ Jm Z Jm ⊆ Jn TJ Z TPMP TJ m ≤ n Z n ≤ m TJ Z TPMP TJ m = n JJJ 6O DPOKVOUP A TF EJDF àOJUP TJ FT WBDÎP P FT FRVJQPUFOUF B BMHÙO TFHNFOUP JOJDJBM EF OBUVSBMFT Jn . JW 6O DPOKVOUP A FT JOàOJUP TJ OP FT àOJUP 0CTFSWBDJPOFT  4J VO DPOKVOUP A FT àOJUP Z OP WBDÎP QPEFNPT FTDSJCJS A = {x1 , . . . , xn }. &O FGFDUP FYJTUFO n ∈ N Z f : Jn → A CJZFDUJWB Z QPS UBOUP TJ QBSB DBEB i ∈ Jn , xi := f (i), FOUPODFT A = {f (i) : 1 ≤ i ≤ n} = {xi : 1 ≤ i ≤ n} ≡ {x1 , . . . , xn }.  &T GÃDJM WFS RVF ∼ FT VOB SFMBDJÓO EF FRVJWBMFODJB FT EFDJS FT SFáFYJWB TJNÊUSJDB Z USBOTJUJWB &KFSDJDJP " %FNVFTUSF RVF MB SFMBDJÓO EF FRVJQPUFODJB ∼ FT VOB SFMBDJÓO EF FRVJWBMFODJB &M TJHVJFOUF MFNB FT EF HSBO JNQPSUBODJB Z OPT TFSÃ EF BZVEB NÃT BEFMBOUF -FNB " 4FBO m ≥ 1 Z n ≥ 1 OÙNFSPT OBUVSBMFT 4J FYJTUF f : Jm → Jn VOP B VOP FOUPODFT m ≤ n : SFDÎQSPDBNFOUF %FNPTUSBDJÓO 'JKBSFNPT MB WBSJBCMF n ∈ N Z VTBSFNPT JOEVDDJÓO TPCSF m &O FM DBTP QBSUJDVMBS n = 1 TJ FYJTUF f : Jm → {1} JOZFDUJWB FOUPODFT f FT VOB GVODJÓO DPOTUBOUF Z QPS MB JOZFDUJWJEBE TF UJFOF RVF m = 1 1PS UBOUP QPEFNPT TVQPOFS RVF n > 1 4FB Pm : FYJTUF

f : Jm → Jn

JOZFDUJWB

1SPCFNPT RVF Pm TF DVNQMF QBSB UPEP m ∈ N.



⇒

m ≤ n.

"QÊOEJDF

4J m = 1 DPNP FM DPOTFDVFOUF EF P1 FT WFSEBEFSP FOUPODFT P1 FT DJFSUB 4VQPOHBNPT BIPSB RVF Pk FT DJFSUB QBSB k ≥ 1 Z EFNPTUSFNPT RVF Pk+1 UBNCJÊO FT DJFSUB 4VQPOHBNPT RVF FYJTUF F : Jk+1 → Jn JOZFDUJWB Z WFBNPT RVF k + 1 ≤ n 1BSB FMMP DPOTJEFSFNPT EPT DBTPT  ∃ l ∈ Jk+1 UBM RVF F (l) = n 1PS TFS F JOZFDUJWB FTUF l FT ÙOJDP 4J l = k + 1 QPS FM TVQVFTUP RVF FTUBNPT IBDJFOEP MB SFTUSJDDJÓO EF F B Jk Z DPO WBMPSFT FO Jn  {n} FT EFDJS MB BQMJDBDJÓO F |Jk : Jk → Jn−1 FT JOZFDUJWB -B IJQÓUFTJT EF JOEVDDJÓO JNQMJDB RVF k ≤ n − 1 Z QPS UBOUP k + 1 ≤ n. "IPSB CJFO TJ l = k + 1 FOUPODFT EFàOJFOEP F (k + 1) := p (p < n) DPOTUSVJNPT MB GVODJÓO φ : Jk → Jn−1 BTÎ F (x) TJ x = l φ(x) := p TJ x = l. &T DMBSP RVF FTUB BQMJDBDJÓO SFTVMUB JOZFDUJWB -B IJQÓUFTJT JOEVDUJWB JNQMJDB RVF k ≤ n − 1  ∀ l ∈ Jk+1 , F (l) = n. %FàOBNPT ψ : Jk+1 → Jn QPS F (x) TJ x = k + 1 ψ(x) := n TJ x = k + 1. $PNP ψ FT JOZFDUJWB FOUPODFT MB BQMJDBDJÓO ψ|Jk : Jk → Jn  {n} UBN CJÊO FT JOZFDUJWB Z FO DPOTFDVFODJB MB IJQÓUFTJT EF JOEVDDJÓO OVFWB NFOUF JNQMJDB RVF k ≤ n − 1. -VFHP FM QSJODJQJP EF JOEVDDJÓO HBSBOUJ[B RVF Pm TF DVNQMF QBSB UPEP m ≥ 1. 1BSB FM SFDÎQSPDP OÓUFTF RVF MB  BQMJDBDJÓO JODMVTJÓO i : Jm → Jn EBEB QPS i(x) = x FT JOZFDUJWB  4VQPOHBNPT RVF MB SFMBDJÓO EF FRVJQPUFODJB ∼ FTUÃ EFàOJEB FO BMHVOB DPMFDDJÓO BEFDVBEB EF DPOKVOUPT C %FOPUBSFNPT QBSB A ∈ C MB DMBTF EF FRVJWBMFODJB EF A QPS |A| Z MB MMBNBNPT DBSEJOBM EFM DPOKVOUP " FT EFDJS |A| := {X ∈ C : X ∼ A}. &O FM DBTP EF RVF FM DPOKVOUP A TFB àOJUP UFOFNPT VOB EFàOJDJÓO BM UFSOBUJWB EFM DBSEJOBM EF A DPNP TJHVF n TJ A ∼ Jn , QBSB BMHÙO n ≥ 1 |A| := 0 TJ A = ∅.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&T FWJEFOUF RVF Jn UJFOF DBSEJOBM n 0CTFSWFNPT RVF |A| FTUÃ CJFO EF àOJEP QVFT TJ A ∼ Jn Z A ∼ Jm FOUPODFT Jn ∼ Jm Z FO WJSUVE EFM -FNB " TF UJFOF m = n /PUFNPT UBNCJÊO RVF EF FTUB EFàOJDJÓO TF EFTQSFOEF RVF FO FM DBTP A àOJUP FM DBSEJOBM EF A FT FM OÙNFSP EF FMFNFOUPT EFM DPOKVOUP A &KFNQMP " &M DPOKVOUP EF MPT OÙNFSPT OBUVSBMFT FT JOàOJUP 4J N GVFSB àOJUP FYJTUJSÎB VOB GVODJÓO f : Jn → N CJZFDUJWB QBSB BMHÙO n ∈ N 1FSP p := f (1) + · · · + f (n) > f (x) QBSB UPEP x ∈ Jn Z BTÎ p∈ / f (Jn ) = N MP DVBM DPOUSBEJDF RVF p ∈ N 0USB GPSNB EF QSPCBS RVF N FT JOàOJUP DPOTJTUF FO BQMJDBS FM BYJPNB EF 1FBOP "1  TVQPOHBNPT RVF N FT àOJUP Z TFB k FM NÃYJNP EF TVT FMFNFOUPT 1PS FM DJUBEP BYJPNB k + FT VO OÙNFSP OBUVSBM Z FT UBM RVF k + > k, MP RVF DPOUSBEJDF MB NBYJNBMJEBE EF k. &KFNQMP " 4FBO A, B, C, D TVCDPOKVOUPT OP WBDÎPT EF VO DPOKVO UP U  4J A ∼ C Z B ∼ D FOUPODFT A × B ∼ C × D. 4J BEFNÃT A ∩ B = C ∩ D = ∅ FOUPODFT A ∪ B ∼ C ∪ D 1SPCFNPT MB QSJNFSB BàSNBDJÓO FYJTUFO CJZFDDJPOFT f : A → C Z g : B →D %FàOBNPT MB GVODJÓO h : A × B → C × D QPS h(a, b) := f (a), g(b) . $PNP f Z g TPO CJZFDUJWBT h UBNCJÊO FT CJZFDUJWB -VFHP A × B ∼ C × D. 1BSB QSPCBS MB TFHVOEB BàSNBDJÓO EFàOBNPT Φ : A ∪ B → C ∪ D QPS f (x) TJ x ∈ A Φ(x) := g(x) TJ x ∈ B. &O QSJNFS MVHBS PCTFSWFNPT RVF Φ FTUÃ CJFO EFàOJEB QVFTUP RVF A ∩ B FT WBDÎP 7FBNPT RVF Φ FT VOP B VOP 4VQPOHBNPT RVF x, y ∈ A ∪ B TPO UBMFT RVF x = y. &OUPODFT TJ x, y ∈ A QPS MB JOZFDUJWJEBE EF f Φ(x) = Φ(y) %F NBOFSB TJNJMBS TF DPODMVZF TJ x, y ∈ B. &O FM DBTP EF RVF x ∈ A Z y ∈ B FT DMBSP RVF Φ(x) = Φ(y) QVFT C ∩ D = ∅. -VFHP Φ FT JOZFDUJWB 'JOBMNFOUF TJ z ∈ C ∪D FOUPODFT MB EJTZVOUJWJEBE EF FTUPT DPOKVOUPT JNQMJDB RVF P CJFO z ∈ C P CJFO z ∈ D &O DVBMRVJFS DBTP MB TPCSFZFDUJWJEBE EF f Z EF g OPT QFSNJUF DPODMVJS RVF FYJTUF x ∈ A ∪ B UBM RVF Φ(x) = z. -VFHP Φ FT TPCSF -FNB " 1BSB UPEP n ∈ N DPO n ≥ 1 UPEP TVCDPOKVOUP EF Jn FT àOJUP %FNPTUSBDJÓO -B EFNPTUSBDJÓO FT QPS JOEVDDJÓO TPCSF n. 4J n = 1 MPT ÙOJDPT TVCDPOKVOUPT EF J1 TPO ∅ Z J1 RVF TPO àOJUPT 4VQPOHBNPT RVF MB BàSNBDJÓO TF DVNQMF QBSB n ≥ 1 Z TFB B ⊆ Jn+1 DPO B = ∅. 1SPCFNPT RVF B FT àOJUP 4J B ⊆ Jn MB IJQÓUFTJT EF JOEVDDJÓO JNQMJDB RVF B FT



"QÊOEJDF

àOJUP 4J B OP FT TVCDPOKVOUP EF Jn FOUPODFT FYJTUF x ∈ B DPO x ∈ / Jn MP RVF JNQMJDB RVF x ∈ Jn+1  Jn . 1PS UBOUP x = n + 1 Z BTÎ n + 1 ∈ B "IPSB CJFO DPNP B ⊆ Jn+1 FYJTUF A ⊆ Jn UBM RVF B = A ∪ {n + 1}. 1PS MB IJQÓUFTJT EF JOEVDDJÓO A FT àOJUP Z QPS UBOUP FYJTUFO r ∈ N Z f : A → Jr CJZFDUJWB %FàOBNPT h : B → Jr+1 QPS f (x) TJ x ∈ A h(x) := r + 1 TJ x = n + 1. %FKBNPT DPNP FKFSDJDJP QBSB FM MFDUPS QSPCBS RVF h FT CJZFDUJWB -VFHP  B ⊆ Jn+1 FT àOJUP Z RVFEB EFNPTUSBEP FM MFNB QBSB UPEP n ≥ 1  &KFSDJDJP " %FNVFTUSF RVF MB BQMJDBDJÓO h EFàOJEB FO MB QSVFCB EFM -FNB " FT CJZFDUJWB 1SPQPTJDJÓO " 4J A FT àOJUP Z B ⊆ A FOUPODFT B FT àOJUP %FNPTUSBDJÓO 4J B FT WBDÎP MB DPODMVTJÓO FT JONFEJBUB 4VQPOHBNPT RVF B = ∅ 1PS MB IJQÓUFTJT FYJTUFO n ∈ N Z f : A → Jn CJZFDUJWB Z FO DPOTFDVFODJB f (B) ⊆ f (A) = Jn  1PS FM MFNB BOUFSJPS f (B) SFTVMUB àOJUP $PNP f |B : B → f (B) FT CJZFDUJWB FOUPODFT f (B) ∼ B Z QPS  UBOUP B FT àOJUP  &T JONFEJBUP MP TJHVJFOUF $PSPMBSJP " 4J A FT JOàOJUP Z B ⊇ A FOUPODFT B UBNCJÊO FT JOà OJUP &KFNQMP " $PNP N ⊂ Z Z N FT JOàOJUP FOUPODFT FM $PSPMBSJP " JNQMJDB RVF Z FT JOàOJUP 5FPSFNB " 4J A Z B TPO àOJUPT FOUPODFT A ∪ B FT VO DPOKVOUP àOJUP Z BEFNÃT |A ∪ B| = |A| + |B| − |A ∩ B|.

"

%FNPTUSBDJÓO 4J BMHVOP EF MPT DPOKVOUPT FT WBDÎP MBT DPODMVTJPOFT TF DVNQMFO USJWJBMNFOUF 4VQPOHBNPT FOUPODFT RVF BNCPT DPOKVOUPT TPO OP WBDÎPT Z RVF BEFNÃT TPO EJTKVOUPT 1PS MB IJQÓUFTJT FYJTUFO OBUVSBMFT m Z n Z GVODJPOFT f : B → Jm , g : A → Jn CJZFDUJWBT "IPSB CJFO VTBOEP MB CJZFDDJÓO h : B → {n + 1, . . . , n + m} EBEB QPS h(k) = f (k) + n,



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

TF EFEVDF RVF B ∼ {n + 1, . . . , n + m} Z BQMJDBOEP FM &KFNQMP " TF UJFOF RVF A ∪ B ∼ Jn ∪ {n + 1, . . . , n + m} = {1, . . . , n + m} = Jn+m . &O DPOTFDVFODJB A ∪ B FT àOJUP "EFNÃT EF MB EFàOJDJÓO EF DBSEJOBM EF VO DPOKVOUP àOJUP |A ∪ B| = n + m = |A| + |B|; FT EFDJS TF DVNQMF " QVFT |A ∩ B| = 0. &O FM DBTP HFOFSBM A ∩ B = ∅, BQMJDBNPT MP PCUFOJEP BOUFSJPSNFOUF B MPT DPOKVOUPT A  B Z B MPT DVBMFT TPO EJTKVOUPT àOJUPT Z A ∪ B = (A  B) ∪ B QPS UBOUP |A ∪ B| = |A  B| + |B|. 1FSP A  B Z A ∩ B TPO DPOKVOUPT àOJUPT Z EJTKVOUPT DVZB VOJÓO FT A Z BTÎ |A| = |A  B| + |A ∩ B| %F FTUBT EPT  ÙMUJNBT JHVBMEBEFT TF EFEVDF "   6TBOEP JOEVDDJÓO TPCSF FM OÙNFSP EF DPOKVOUPT TF EFNVFTUSB RVF MB VOJÓO EF VO OÙNFSP àOJUP EF DPOKVOUPT àOJUPT FT VO DPOKVOUP àOJUP $PSPMBSJP " 4FB Ai àOJUP QBSB i = 1, 2, . . . , k &OUPODFT

k (

Ai FT

i=1

VO DPOKVOUP àOJUP

" &KFSDJDJPT  %FNVFTUSF FM $PSPMBSJP "  1SVFCF RVF TJ |A| = n ∈ N FOUPODFT |P(A)| = 2n , EPOEF P(A) EFOPUB FM DPOKVOUP EF QBSUFT EF A.  4J A FT VO DPOKVOUP àOJUP EF n FMFNFOUPT Z B FT VO DPOKVOUP àOJUP EF k FMFNFOUPT yDVÃOUBT GVODJPOFT EF A FO B TF QVFEFO EFàOJS y$VÃOUBT GVODJPOFT JOZFDUJWBT EF A FO B TF QVFEFO FTUBCMFDFS  4FB k ∈ N àKP %FNVFTUSF RVF N ∼ N  {k}  4FB A = ∅ 1SVFCF RVF A × {x} ∼ {x} × A ∼ A.  4FBO A1 , . . . , Ak DPOKVOUPT àOJUPT NVUVBNFOUF EJTKVOUPT FT EFDJS TJ j = i FOUPODFT Aj ∩ Ai = ∅ %FNVFTUSF RVF k k (     Ai  = |Ai |.  i=1



i=1

"QÊOEJDF

 4FBO A Z B DPOKVOUPT àOJUPT &OUPODFT A × B FT àOJUP Z BEFNÃT |A × B| = |A||B| 4VHFSFODJB VTF MPT FKFSDJDJPT  Z 

 4FBO A Z B DPOKVOUPT àOJUPT DPO A ⊆ B 1SVFCF RVF |A| ≤ |B|. y&T DJFSUP FM SFDÎQSPDP  4FBO A, B Z C DPOKVOUPT àOJUPT 1SVFCF RVF |A∪B∪C| = |A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|.  4FB A = ∅ VO DPOKVOUP àOJUP %FNVFTUSF RVF FYJTUF VOB GVODJÓO JOZFDUJWB f : A → N y&T DJFSUP FM SFDÎQSPDP  4FB A = ∅ VO DPOKVOUP àOJUP %FNVFTUSF RVF FYJTUF VOB GVODJÓO g : N → A TPCSFZFDUJWB y&T DJFSUP FM SFDÎQSPDP  4J f : A → B FT JOZFDUJWB Z A FT JOàOJUP QSVFCF RVF B FT JOàOJUP  %FNVFTUSF RVF TJ A  B ∼ B  A FOUPODFT A ∼ B  4J A FT JOàOJUP Z B ⊂ A FT àOJUP FOUPODFT A  B FT JOàOJUP

" .ÃT TPCSF DBNQPT &O FTUB TFDDJÓO TF QSFTFOUBO EF NBOFSB CSFWF BMHVOPT IFDIPT SFMBDJP OBEPT DPO FM DPODFQUP EF DBNQP *OJDJBNPT DPO BMHVOPT FKFNQMPT &KFNQMP " 4FB F = {0, 1, 2, 3} Z EFàOBNPT FO F MBT PQFSBDJPOFT ⊕ Z  EF MB TJHVJFOUF NBOFSB m⊕n = m + n Z mn = mn EPOEF m TJHOJàDB FM SFTUP BM EJWJEJS m QPS 4 &M BMHPSJUNP EF MB EJWJTJÓO JNQMJDB RVF ⊕ Z  TPO PQFSBDJPOFT CJOBSJBT -BT TJHVJFOUFT UBCMBT JMVTUSBO UPEPT MPT SFTVMUBEPT ⊕    

    

    

    

    

    

    

    

    

    

&KFNQMP " 4FB F FM DPOKVOUP EFM FKFNQMP BOUFSJPS EPUBEP DPO MBT NJTNBT PQFSBDJPOFT ⊕ Z  &T DMBSP RVF 0 FT OFVUSP QBSB MB TVNB FO DÎSDVMP ⊕ Z RVF 1 FT OFVUSP QBSB FM QSPEVDUP FO DÎSDVMP  y&T F, ⊕,  VO DBNQP /P QPSRVF 2 ∈ F FT EJTUJOUP EF 0 Z OP FYJTUF b ∈ F WFS UBCMB EF  UBM RVF 2  b = 1 FT EFDJS OP TF DVNQMF FM BYJPNB 1* 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&YJTUF VOB NBOFSB EF PCUFOFS DBNQPT àOJUPT DPNP TF JMVTUSB FO FM FKFNQMP RVF TJHVF &KFNQMP " 4FBO p ∈ N QSJNP Z Zp := {0, 1, . . . , p − 1} EPUBEP EF MBT PQFSBDJPOFT CJOBSJBT ⊕ Z  NFODJPOBEBT FO MPT FKFNQMPT BOUFSJPSFT 7FBNPT RVF  Zp , ⊕,  FT VO DBNQP &O FM DBTP QBSUJDVMBS EF VO WBMPS EF p QPS FKFNQMP p = 3 FODPOUSBNPT RVF QBSB QSPCBS TPMBNFOUF 4"

FM BYJPNB EF MB BTPDJBUJWJEBE EF ⊕ OFDFTJUBNPT WFSJàDBS 33 JHVBMEBEFT MP DVBM FT QPDP QSÃDUJDP y2VÊ IBDFS FOUPODFT QBSB QSPCBS RVF ⊕ Z  DVNQMFO MPT BYJPNBT EF DBNQP &YJTUF VO DPODFQUP EF MB SBNB EFM ÃMHF CSB RVF OPT TFSÃ EF NVDIB VUJMJEBE QBSB MMFWBS B DBCP OVFTUSB QSVFCB IPNPNPSàTNP (SPTTP NPEP FTUF UÊSNJOP EFTDSJCF VOB BQMJDBDJÓO EF VO TJTUFNB BMHFCSBJDP FO PUSP RVF QSFTFSWB MB FTUSVDUVSB 6OB DBSBDUFSÎTUJDB EF MPT IPNPNPSàTNPT FT RVF DVBOEP TF DPOPDF VO DÓNQVUP FO VOP EF MPT TJTUFNBT TF QVFEF SFBMJ[BS FM DÓNQVUP DPO MPT FMFNFOUPT DPSSFTQPO EJFOUFT FO FM PUSP &T FTB MB JEFB RVF WBNPT B VUJMJ[BS 4FB h : Z → Zp EBEB QPS h(m) := m EPOEF m TJHOJàDB FM SFTUP PCUF OJEP BM EJWJEJS m QPS p &M BMHPSJUNP EF MB EJWJTJÓO HBSBOUJ[B RVF h FTUÃ CJFO EFàOJEB -B JEFB FT VTBS MB GVODJÓO h QBSB MMFWBS MBT QSPQJFEBEFT EF MB TVNB PSEJOBSJB FO Z B MB TVNB ⊕ EFàOJEB FO Zp  1BSB FTUP TF EFNVFTUSB RVF h FT BEJUJWB FT EFDJS h(m + n) = h(m) ⊕ h(n) OPUFNPT QSJNFSP RVF h(x) = x QBSB UPEP x ∈ Zp FO QBSUJDVMBS h FT TPCSFZFDUJ WB  1BSB m, n ∈ Z FM BMHPSJUNP EF MB EJWJTJÓO EJDF RVF FYJTUFO ÙOJDPT FOUFSPT q, q  , r Z r DPO 0 ≤ r < p Z 0 ≤ r < p UBMFT RVF m = pq + r = pq + m ,

n = pq  + r = pq  + n.

"

"IPSB CJFO QPS EFàOJDJÓO EF ⊕ UFOFNPT RVF h(m)⊕h(n) = h(m) + h(n) Z QPS PUSB QBSUF EF "

TF EFEVDF RVF h(m) + h(n) = m + n = m + n − p(q + q  ).

"

0CTFSWFNPT RVF TJ c ∈ Z FOUPODFT QBSB UPEP z ∈ Z TF UJFOF RVF h(z + cp) = h(z)  &O FGFDUP OVFWBNFOUF FM BMHPSJUNP EF MB EJWJTJÓO HBSBOUJ[B RVF z = c p + r DPO 0 ≤ r < p Z QPS UBOUP z + cp = c p + r + cp = (c + c)p + r ; EF MB EFàOJDJÓO EF h TF TJHVF RVF h(z + cp) = r = h(z). -VFHP EF "

TF UJFOF RVF h(m)⊕h(n) = h(m) + h(n) = m + n − p(q + q  ) = m + n = h(m+n).





&TUP TJHOJàDB RVF h FT QFSJÓEJDB DPO QFSÎPEP p.

"QÊOEJDF

6O SB[POBNJFOUP BOÃMPHP NVFTUSB RVF h FT NVMUJQMJDBUJWB FTUP FT h(mn) = h(m)  h(n). $PO BZVEB EF FTUBT EPT QSPQJFEBEFT EF MB GVODJÓO h TF QSVFCBO MPT BYJPNBT EF DBNQP QBSB Zp  7FBNPT RVF ⊕ FT BTPDJBUJWB TFBO m, n Z k FO Zp  1PS MB TPCSFZFDUJWJEBE EF h FYJTUFO m , n Z k  FO Z UBMFT RVF h(m ) = m, h(n ) = n Z h(k  ) = k -B DMBWF FT BQSPWFDIBS MB BTPDJBUJWJEBE EF + FO Z Z MBT QSPQJFEBEFT EF h QBSB USBOTGFSJS FTUB NJTNB QSPQJFEBE B ⊕ FO Zp BTÎ (m ⊕ n) ⊕ k = [h(m ) ⊕ h(n )] ⊕ h(k  ) = h(m + n ) ⊕ h(k  ) = h[(m + n ) + k  ] = h[m + (n + k  )] = h(m ) ⊕ h(n + k  ) = h(m ) ⊕ [h(n ) ⊕ h(k  )] = m ⊕ (n ⊕ k). %F NBOFSB TJNJMBS TF QSVFCBO 1"

4$

1$ Z %  $MBSBNFOUF 0 FT OFVUSP EF ⊕ Z FM TJHVJFOUF DÓNQVUP NVFTUSB RVF 1 FT OFVUSP QBSB  TJ m ∈ Zp FOUPODFT m = h(m) = h(m · 1) = h(m)  h(1) = m  1. 1BSB QSPCBS MB FYJTUFODJB EF JOWFSTPT QBSB ⊕ OÓUFTF RVF TJ m ∈ Zp DPO m > 0 FOUPODFT 0 < p − m < p FT EFDJS p − m ∈ Zp  "EFNÃT m ⊕ (p − m) = m + (p − m) = p = h(p) = 0, QFSNJUF DPODMVJS RVF p − m FT JOWFSTP BEJUJWP EF m 'JOBMNFOUF WFB NPT MB FYJTUFODJB EF JOWFSTPT QBSB  TFB m = 0, m ∈ Zp  &O FTUF QVOUP VTBSFNPT MB IJQÓUFTJT EF MB QSJNBMJEBE EF p $PNP m < p TV .ÃYJNP $PNÙO %JWJTPS FT 1 Z QPS UBOUP FYJTUFO FOUFSPT x, y UBMFT RVF mx + py = 1 -VFHP h(mx + py) = 1 = h(mx) = h(m)  h(x) = m  h(x). &T DMBSP RVF h(x) = 0 Z QPS UBOUP m−1 = h(x) ∈ Zp  2VFEB DPNQMFUB MB QSVFCB EF RVF  Zp , ⊕,   FT VO DBNQP DVBOEP p FT QSJNP &KFSDJDJP " &O FM DBNQP  Z5 , ⊕,   FODVFOUSF

1 1 3 2 , , , · 2 4 2 4

&KFNQMP " 4FB F VO DBNQP FO FM DVBM 1 + 1 = 0 1PS FKFNQMP  Z2 , ⊕,   FT VOP EF FMMPT %FNVFTUSF RVF QBSB UPEP a ∈ F, a + a = 0 &O FGFDUP a + a = 1 · a + 1 · a = (1 + 1)a = 0 · a = 0



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP " %FNPTUSBS RVF FO VO DBNQP F TF DVNQMF    ∀ m, n ∈ N∗ (1 + 1 + · · · + 1) · (1 + 1 + · · · + 1) = 1 + 1 + · · · + 1         m WFDFT

mn WFDFT

n WFDFT

-B QSVFCB TF IBDF QPS JOEVDDJÓO TPCSF m Z QBSB TJNQMJàDBS MB FTDSJ UVSB VTBSFNPT MB OPUBDJÓO n · 1 FO MVHBS EF 1 + 1 + · · · + 1  n WFDFT

4J m = 1 QPS FM BYJPNB 1/

MB BàSNBDJÓO FT DJFSUB 4VQPOHBNPT RVF QBSB m = k ≥ 1 TF DVNQMF MB BàSNBDJÓO Z QSPCFNPT QBSB m = k + 1 "QMJDBOEP MB BTPDJBUJWJEBE EF MB TVNB VO OÙNFSP àOJUP EF WFDFT %

1/ Z MB IJQÓUFTJT JOEVDUJWB UFOFNPT (1 + 1 + · · · + 1) (1 + 1 + · · · + 1) = [(k + 1) · 1][n · 1]      n WFDFT

k+1 WFDFT

= [k · 1 + 1][n · 1] = [k · 1][n · 1] + n · 1 = (kn) · 1 + n · 1 = (kn + n) · 1 = [(k + 1)n] · 1 = (1 + 1 + · · · + 1) .    (k+1)n WFDFT

-VFHP MB BàSNBDJÓO TF UJFOF QBSB UPEP m ∈ N &KFNQMP " 4FB F VO DBNQP Z TVQÓOHBTF RVF QBSB BMHÙO OBUVSBM k TF UJFOF 1 + 1 + "

· · · + 1 = 0. k WFDFT

%FNPTUSBS RVF TJ p FT FM NFOPS OBUVSBM DPO FTUB QSPQJFEBE FOUPODFT p FT VO OÙNFSP QSJNP &O FGFDUP TFB p FM NFOPS OBUVSBM RVF WFSJàDB " Z TVQPOHBNPT RVF OP FT QSJNP &OUPODFT FYJTUFO OBUVSBMFT m Z n DPO 1 < m < p Z 1 < n < p UBMFT RVF p = mn 1PS FM FKFNQMP BOUFSJPS 0 = 1 + 1 + · · · + 1 = 1 + 1 + · · · + 1 p WFDFT

mn WFDFT

= (1 + 1 + · · · + 1) (1 + 1 + · · · + 1) .      m WFDFT

n WFDFT

-B DPOTFDVFODJB $ EF MPT BYJPNBT EF DBNQP 5FPSFNB  QÃH 

JNQMJDB RVF 1 + 1 + · · · + 1 = 0 m WFDFT



Ó

1 + 1 + · · · + 1 = 0. n WFDFT

"QÊOEJDF

$VBMRVJFSB EF MBT EPT BMUFSOBUJWBT DPOUSBEJDF MB NJOJNBMJEBE EF p. -VF HP p FT QSJNP &O FM FKFNQMP BOUFSJPS FM NFOPS OBUVSBM p RVF DVNQMF " TF MMBNB DB SBDUFSÎTUJDB EFM DBNQP F &O FM DBTP EF RVF OP FYJTUB UBM OBUVSBM EFDJNPT RVF MB DBSBDUFSÎTUJDB FT DFSP &KFNQMP " $POTJEFSFNPT FM DBNQP  Zp , ⊕,  FTUVEJBEP FO FM &KFN QMP " &TUF DBNQP OP FT PSEFOBEP QVFT TJ MP GVFSB FM BYJPNB EF DFSSBEVSB JNQMJDBSÎB RVF 0 = 1 ⊕ 1 ⊕ · · · ⊕ 1 ∈ P. p WFDFT

&O SFBMJEBE TJ VO DBNQP FT àOJUP FOUPODFT OP QVFEF TFS PSEFOBEP &M TJHVJFOUF UFPSFNB FTUBCMFDF FTUB BàSNBDJÓO 5FPSFNB " 4J F FT VO DBNQP PSEFOBEP FOUPODFT F FT JOàOJUP %FNPTUSBDJÓO 4F EFàOF MB GVODJÓO ϕ : N∗ → F QPS ϕ(n) := 1 + 1 + · · · + 1 . n WFDFT

7FBNPT RVF ϕ SFTVMUB JOZFDUJWB &GFDUJWBNFOUF TJ ϕ(m) = ϕ(n) Z m = n TJO QFSEFS HFOFSBMJEBE QPEFNPT TVQPOFS m < n

FOUPODFT 1 + 1 + · · · + 1 = 1 + 1 + · · · + 1 m WFDFT

n WFDFT

Z TFHÙO MB QSPQJFEBE DBODFMBUJWB 1 + 1 + · · · + 1 = 0, n−m WFDFT

DPO n − m ∈ N∗  &TUP OP QVFEF PDVSSJS QVFT F FT PSEFOBEP 1PS UBOUP m = n Z BTÎ ϕ FT VOP B VOP &T DMBSP RVF ϕ(N∗ ) ⊆ F Z RVF ϕ : N∗ → ϕ(N∗ ) FT CJZFDUJWB FO DPOTFDVFODJB ϕ(N∗ ) ∼ N∗  $PNP N∗ FT JOàOJUP ϕ(N∗ )  UBNCJÊO MP FT Z FTUP JNQMJDB RVF F FT JOàOJUP 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

" 1PMJOPNJPT &O FTUB TFDDJÓO QSFTFOUBNPT BMHVOPT UFNBT FMFNFOUBMFT TPCSF QPMJOP NJPT DPO MB JOUFODJÓO EF QSPWFFS BM MFDUPS JOUFSFTBEP MB JOGPSNBDJÓO NÎOJNB SFRVFSJEB TPCSF FTUF QBSUJDVMBS &TUÃ FODBNJOBEP B VOB QSFTFO UBDJÓO EF UJQP PQFSBUJWP Z QPS UBOUP MBT QSVFCBT TFSÃO QPDBT Z MBT SFNJ UJNPT B SFGFSFODJBT P DVSTPT TVQFSJPSFT EF ÃMHFCSB &YJTUF VOB NBOFSB EF QSFTFOUBS FM DPODFQUP EF QPMJOPNJP FM DVBM DPOTJTUF FO VTBS TVDFTJPOFT DVZPT UÊSNJOPT EF VO DBNQP K TPO OVMPT FYDFQUP VO OÙNFSP àOJUP EF FMMPT QFSP MB QSFTFOUBDJÓO RVF IBDFNPT FO FTUF UFYUP QPS SB[POFT EF TJNQMJDJEBE TFSÃ MB USBEJDJPOBM %FàOJDJÓO " 4FB K VO DBNQP 1PS VO QPMJOPNJP TPCSF K P DPO DPFàDJFOUFT FO K FOUFOEFNPT VOB FYQSFTJÓO GPSNBM RVF UJFOF MB GPSNB p(x) = an xn + an−1 xn−1 + · · · + a0 , EPOEF n ∈ N DBEB DPFàDJFOUF ai ∈ K Z x FT VOB WBSJBCMF P JOEFUFSNJ OBEB 4J an = 0 EFDJNPT RVF FM QPMJOPNJP p UJFOF HSBEP n MP DVBM EFOPUB NPT QPS ;`/ (p) = n 1PS SB[POFT UÊDOJDBT FM HSBEP EFM QPMJOPNJP DFSP 0xn + 0xn−1 + · · · + 0, TF EFàOF DPNP −∞. %FOPUBNPT QPS K[x] FM DPOKVOUP EF QPMJOPNJPT DPO DPFàDJFOUFT FO K &O EJDIP DPOKVOUP TF EFàOF MB JHVBMEBE Z MBT PQFSBDJPOFT EF TVNB Z QSPEVDUP EF MB TJHVJFOUF NBOFSB QBSB EPT QPMJOPNJPT p(x) = an xn + an−1 xn−1 + · · · + a0 Z q(x) = bm xm + bm−1 xm−1 + · · · + b0 EFDJNPT RVF p = q TJ ai = bi QBSB UPEP i TJ QPS FKFNQMP m < n FOUPODFT am+1 = · · · = an = 0  -B TVNB p + q TF EFàOF DPNP p(x) + q(x) := a0 + b0 + (a1 + b1 )x + · · · + (aN + bN )xN , EPOEF N = NÃY{n, m}. 'JOBMNFOUF FM QSPEVDUP pq TF EFàOF DPNP p(x)q(x) := a0 b0 + (a0 b1 + a1 b0 )x + (a0 b2 + a1 b1 + a2 b0 )x2 + · · · + ck xk + · · · + an bm xn+m , EPOEF ck =

k 

aj bk−j .

j=0

%BEP RVF FO HFOFSBM m = n TJO FNCBSHP QPEFNPT TVQPOFS RVF m = n ZB RVF MPT DPFàDJFOUFT GBMUBOUFT FO BMHVOP EF MPT QPMJOPNJPT TF UPNBO JHVBMFT B DFSP





7ÊBTF QPS FKFNQMP QÃH  P 

"QÊOEJDF

&O MP RVF SFTUB EF FTUB TFDDJÓO USBCBKBSFNPT TPMP DPO MPT DBNQPT K = R, K = Q P K = C, B NFOPT RVF TF FTQFDJàRVF PUSP -B SB[ÓO UÊDOJDB JOUSPEVDJEB FO MB BOUFSJPS EFàOJDJÓO MB DPNQMFNFO UBNPT DPO MBT DPOWFODJPOFT −∞ + (−∞) = −∞;

−∞ + k = −∞ Z

− ∞ < k,

∀ k ∈ N.

"TÎ TJO FYDFQDJPOFT UJFOFO WBMJEF[ MPT TJHVJFOUFT SFTVMUBEPT &KFSDJDJP " 4FBO p Z q QPMJOPNJPT %FNVFTUSF RVF ;`/ (p + q) ≤ NÃY{;`/ (p), ;`/ (q)}. 5FPSFNB " 4FBO p(x) = an xn + an−1 xn−1 + · · · + a0 Z q(x) = bm xm + bm−1 xm−1 + · · · + b0 QPMJOPNJPT &OUPODFT ;`/ (pq) = ;`/ (p) + ;`/ (q). %FNPTUSBDJÓO 4J BMHVOP EF MPT QPMJOPNJPT FT DFSP QPS MBT DPOWFODJP OFT FTUBCMFDJEBT MB DPODMVTJÓO TF UJFOF JONFEJBUBNFOUF 4VQPOHBNPT FOUPODFT RVF BNCPT QPMJOPNJPT TPO OP OVMPT Z TPO UBMFT RVF ;`/ (p) = n Z ;`/ (q) = m &TUP JNQMJDB RVF an = 0 Z bm = 0. -VFHP EF MB EFàOJ DJÓO EF QSPEVDUP EF QPMJOPNJPT TF UJFOF RVF ;`/ (pq) = n + m = ;`/ (p) + ;`/ (q), DPO MP DVBM TF DPNQMFUB MB QSVFCB

 

0CTFSWBDJÓO 4J MPT DPFàDJFOUFT OP QFSUFOFDFO B VO DBNQP TJOP B VO BOJMMP FM UFPSFNB BOUFSJPS OP FT DJFSUP 1PS FKFNQMP DPOTJEFSFNPT MPT DPFàDJFOUFT FO FM BOJMMP Z6 Z MPT QPMJOPNJPT p(x) = 2x2 − 1, q(x) = 3x + 2. &OUPODFT ;`/ (p) + ;`/ (q) = 3 NJFOUSBT RVF FM QPMJOPNJP pq = 6x3 + 4x2 − 3x − 2 = 4x2 + 3x + 4 UJFOF HSBEP 2 6O QBS EF DPOTFDVFODJBT TF EFTQSFOEFO EF FTUF UFPSFNB $PSPMBSJP " 4J p Z q TPO QPMJOPNJPT UBMFT RVF pq = 0 FOUPODFT BMHVOP EF FMMPT EFCF TFS FM QPMJOPNJP DFSP



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

%FNPTUSBDJÓO 4J BNCPT GVFSBO EJGFSFOUFT EF DFSP Z DPNP ;`/ (pq) = ;`/ (p) + ;`/ (q) FOUPODFT FM MBEP J[RVJFSEP EF FTUB JHVBMEBE TFSÎB −∞ NJFOUSBT RVF FM MBEP EFSFDIP TFSÎB VO OBUVSBM &TUB DPOUSBEJDDJÓO EF  NVFTUSB FM DPSPMBSJP  &O DBTP EF RVF FM QSPEVDUP TFB VO QPMJOPNJP DPOTUBOUF EJGFSFOUF EF DFSP UFOFNPT RVF 0 = ;`/ (pq) = ;`/ (p) + ;`/ (q) MP DVBM JNQMJDB RVF 0 = ;`/ (p) = ;`/ (q) Z BTÎ BNCPT QPMJOPNJPT p Z q TPO DPOTUBOUFT $PO FTUP RVFEB QSPCBEP MP TJHVJFOUF $PSPMBSJP " 4J p Z q TPO QPMJOPNJPT UBMFT RVF TV QSPEVDUP pq FT VO QPMJOPNJP DPOTUBOUF OP OVMP FOUPODFT BNCPT QPMJOPNJPT TPO DPOTUBOUFT %FàOJDJÓO " 4FBO α ∈ K Z p(x) = an xn + an−1 xn−1 + · · · + a0 ∈ K[x]. 4F EFàOF p(α) DPNP p(α) = an αn + an−1 αn−1 + · · · + a0 . %FDJNPT RVF α FT VOB SBÎ[ P VO DFSP EF p(x) TJ p(α) = 0. &KFNQMP " 4FB p(x) = ax2 + bx + c ∈ K[x] UBM RVF p(α) = 0 QBSB UPEP α ∈ K. 7FBNPT RVF p = 0. &O FGFDUP IBDJFOEP α = 0, 1, −1 PCUFOFNPT RVF c = 0, a + b = 0 Z a = b SFTQFDUJWBNFOUF %F FTUP TF EFEVDF RVF a = b = c = 0 Z QPS UBOUP p = 0. &KFNQMP " &M $PSPMBSJP " OP TF DVNQMF TJ FM DBNQP FT àOJUP 1BSB JMVTUSBS FTUP DPOTJEFSFNPT K = Z2 := {0, 1}, p(x) = x + 1 Z q(x) = x. &O FTUB JMVTUSBDJÓO UFOFNPT RVF p(x)q(x) = x2 + x DPO p(x)q(x) = 0 QBSB UPEP x ∈ Z2  &TUP JNQMJDB TFHÙO FM FKFNQMP BOUFSJPS RVF p q = 0 TJO FNCBSHP p = 0 Z q = 0. &M DPOKVOUP K[x] DPO MBT PQFSBDJPOFT EF TVNB Z QSPEVDUP FO ÊM EF àOJEPT OP FT VO DBNQP QVFT MB QSPQJFEBE JOWFSUJWB OP TF TBUJTGBDF TJO FNCBSHP SFTVMUB TFS VO EPNJOJP FOUFSP FM DVBM FT TJNJMBS BM EPNJOJP FOUFSP Z Z FO FTUF IBZ VOB UFPSÎB EF EJWJTJCJMJEBE QPS UBOUP UFOFNPT TV BOÃMPHB FO K[x] &NQF[BNPT FOVODJBOEP FM BMHPSJUNP EF MB EJWJ TJÓO QBSB QPMJOPNJPT DVZB QSVFCB QVFEF DPOTVMUBSTF QPS FKFNQMP FO MB SFGFSFODJB QÃH  5FPSFNB " "MHPSJUNP EF MB EJWJTJÓO  4FBO p, q ∈ K[x] DPO q = 0 &OUPODFT FYJTUFO ÙOJDPT QPMJOPNJPT c(x) FM DPDJFOUF Z r(x) FM SFTUP

FO K[x] UBMFT RVF p(x) = c(x)q(x) + r(x) DPO r = 0 P CJFO ;`/ (r) < ;`/ (q).



"QÊOEJDF

'JKFNPT a ∈ K. $PNP VOB BQMJDBDJÓO JONFEJBUB EFM BMHPSJUNP EF MB EJWJTJÓO DPO q(x) = x − a FYJTUF VO ÙOJDP QPMJOPNJP c(x) Z VO ÙOJDP FMFNFOUP r ∈ K UBMFT RVF p(x) = c(x)(x − a) + r. 0CTFSWFNPT RVF r = p(a). $PO FTUP QPEFNPT FOVODJBS FM TJHVJFOUF SFTVMUBEP 5FPSFNB " 5FPSFNB EFM SFTJEVP  &M SFTUP BM EJWJEJS VO QPMJOPNJP p(x) QPS x − a FT p(a). %F MP BOUFSJPS TF EFTQSFOEF FM TJHVJFOUF SFTVMUBEP CJFO DPOPDJEP 5FPSFNB " 5FPSFNB EFM GBDUPS  4J a FT VO DFSP EF VO QPMJOPNJP p(x) FOUPODFT FYJTUF VO ÙOJDP QPMJOPNJP c(x) UBM RVF p(x) = c(x)(x−a). &KFNQMP " )BMMFNPT FM WBMPS EF MB DPOTUBOUF k ∈ R EF NBOFSB RVF x − 4 TFB GBDUPS EFM QPMJOPNJP p(x) = 9x4 − 35x3 + kx2 + kx − 4. 1BSB FTUP CBTUB IBMMBS k UBM RVF p(4) = 0, FT EFDJS 9 · 44 − 35 · 43 + 20k = 4, MP DVBM TF TBUJTGBDF QBSB k = −3. 4J FO FM ÙMUJNP FKFNQMP RVJTJÊSBNPT FODPOUSBS FM QPMJOPNJP DPDJFOUF c(x) yDÓNP QPEFNPT QSPDFEFS #JFO BEFNÃT EF MB NBOFSB FTUÃOEBS QB SB EJWJEJS QPMJOPNJPT FYJTUFO PUSBT UÊDOJDBT QBSB IBMMBS EJDIP DPDJFOUF 1SFTFOUBNPT B DPOUJOVBDJÓO VOB EF FMMBT MMBNBEB MB SFHMB EF 3VGàOJ "OUFT EF QSFTFOUBS EJDIB SFHMB FO BDDJÓO WFBNPT VO SB[POBNJFOUP RVF MB FYQMJDB TFBO a ∈ R Z p(x) = an xn + an−1 xn−1 + · · · + a0 ∈ R[x] VO QPMJOPNJP EF HSBEP n > 0 4BCFNPT RVF FYJTUFO ÙOJDPT c(x) ∈ R[x] Z r ∈ R UBMFT RVF p(x) = c(x)(x−a)+r. $PNP FT DMBSP RVF ;`/ (c) = n−1 TVQPOHBNPT RVF c(x) = qn−1 xn−1 + qn−2 xn−2 + · · · + q1 x + q0 Z IBMMFNPT MPT DPFàDJFOUFT EF FTUF QPMJOPNJP %F MB JHVBMEBE an xn + an−1 xn−1 + · · · + a0 = (x − a)(qn−1 xn−1 + qn−2 xn−2 + · · · + q1 x + q0 ) + r = qn−1 xn + (qn−2 − aqn−1 )xn−1 + (qn−3 − aqn−2 )xn−2 + · · · + (q0 − aq1 )x + r − aq0 ,



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

PCUFOFNPT RVF an = qn−1

⇐⇒ qn−1 = an

an−1 = qn−2 − aqn−1 ⇐⇒ qn−2 = aqn−1 + an−1 an−2 = qn−3 − aqn−2 ⇐⇒ qn−3   a1 = q0 − aq1

⇐⇒

a0 = r − aq0 .

⇐⇒

= aqn−2 + an−2  

q0 = aq1 + a1 r = aq0 + a0 .

&TUB ÙMUJNB DBEFOB EF JHVBMEBEFT OPT QFSNJUFO SFTVNJS MB 3FHMB EF 3VG àOJ FO FM TJHVJFOUF FTRVFNB $PFàDJFOUFT EF p(x) $PFG EF c(x) Z FM SFTUP

an

an−1 aqn−1

an−2 aqn−2

··· ···

a1 aq1

a0 aq0

qn−1

qn−2

qn−3

···

q0

r

0CTFSWFNPT EPT DPTBT FO FM FTRVFNB BOUFSJPS QSJNFSP MPT UÊSNJOPT EF MB TFHVOEB àMB TF PCUJFOFO NVMUJQMJDBOEP FM OÙNFSP a QPS FM DPFà DJFOUF EF c(x) QSFWJBNFOUF DPOPDJEP FO MB ÙMUJNB àMB

Z TFHVOEP QPS MBT JHVBMEBEFT EF BSSJCB B MB EFSFDIB DBEB DPFàDJFOUF EF c(x) TF PCUJFOF TVNBOEP MB SFTQFDUJWB DPMVNOB &KFNQMP " 1BSB JMVTUSBS MB NFDÃOJDB DPO MB SFHMB EF 3VGàOJ IBMMF NPT FM DPDJFOUF Z FM SFTJEVP BM EJWJEJS FM QPMJOPNJP p(x) = 3x7 + 31x4 − x6 + 21x + 5 QPS x + 2. &O FTUF DBTP a = −2. &TDSJCBNPT FTRVFNÃUJDBNFOUF MB JO GPSNBDJÓO MB QSJNFSB àMB DPOUJFOF MPT DPFàDJFOUFT EFM QPMJOPNJP EBEP UFOJFOEP FM DVJEBEP EF FTDSJCJS MPT DPFàDJFOUFT EF MPT UÊSNJOPT EF NBZPS B NFOPS HSBEP &M QSJNFS UÊSNJOP EF MB TFHVOEB àMB RVF BQBSFDF FO MB TFHVOEB DPMVNOB TF PCUJFOF NVMUJQMJDBOEP a = −2 QPS 3 Z BTÎ MB QSJNFSB DPMVNOB FTUÃ DPOGPSNBEB QPS −1 Z −6 MP DVBM TVNB −7 &TUB TVNB TF NVMUJQMJDB QPS a = −2 Z FM SFTVMUBEP FT FM UÊSNJOP TJHVJFOUF FO MB TFHVOEB àMB &TUF QSPDFTP TF SFQJUF IBTUB UFSNJOBS &O MB UFSDFSB àMB RVFEBO MPT DPFàDJFOUFT EFM QPMJOPNJP DPDJFOUF CVTDBEP FYDFQUP QPS FM ÙMUJNP OÙNFSP RVF SFQSFTFOUB FM SFTUP EF MB EJWJTJÓO $PFàDJFOUFT EF p(x)



−1 −6

 

 −28

 −6

 

 −24

 

$PFG CVTDBEPT Z FM SFTUP

3

−7





−6

12

−3





"QÊOEJDF

%F FTUB NBOFSB FM QPMJOPNJP DPDJFOUF CVTDBEP FT 3x6 − 7x5 + 14x4 + 3x3 − 6x2 + 12x − 3 Z FM SFTUP EF MB EJWJTJÓO FT 11. &M TJHVJFOUF SFTVMUBEP QSPQPSDJPOB DPOEJDJPOFT OFDFTBSJBT QBSB RVF VO SBDJPOBM TFB DFSP EF VO QPMJOPNJP DPO DPFàDJFOUFT SBDJPOBMFT 5FPSFNB " 4FB p(x) = an xn + an−1 xn−1 + · · · + a0 ∈ R[x] EPOEF DBEB ai ∈ Z Z TFB r/s ∈ Q DPO (r, s) = 1 VO DFSP EF p(x). &OUPODFT r | a0 Z s | a n . %FNPTUSBDJÓO $PNP 0 = p(r/s) = an (r/s)n + an−1 (r/s)n−1 + an−2 (r/s)n−2 + · · · + a0 , NVMUJQMJDBOEP QPS sn PCUFOFNPT an rn + an−1 rn−1 s + an−2 rn−2 s2 · · · + a0 sn = 0; EF FTUP TF EFEVDF RVF (an−1 rn−1 + an−2 rn−2 s + · · · + a0 sn−1 )s = −an rn . &TUP JNQMJDB RVF s | an rn Z DPNP (r, sn ) = 1 FOUPODFT s | an . %F NBOFSB   TJNJMBS TF EFNVFTUSB RVF r | a0 . &KFNQMP " 'BDUPSJDFNPT FO FM DBNQP Q TJ FT QPTJCMF FM QPMJOPNJP p(x) = 4x3 − 24x2 + 23x + 18 1BSB FTUP CBTUB RVF FODPOUSFNPT MPT FWFOUVBMFT DFSPT SBDJPOBMFT 4FHÙO FM UFPSFNB BOUFSJPS TJ r/s ∈ Q FT VO DFSP EF p(x) FOUPODFT MBT SBÎDFT DBOEJEBUBT EFCFO TBUJTGBDFS RVF r | 18 Z s | 4. 1PS UBOUP FM DPOKVOUP EF MPT QPTJCMFT DFSPT EF FTUF QPMJOPNJP FT 1 1 3 3 9 9 {±1, ±2, ±3, ±6, ±9, ±18, ± , ± , ± , ± , ± , ± }. 2 4 2 4 2 4 "M FWBMVBS FODPOUSBNPT RVF MPT DFSPT SBDJPOBMFT EF p(x) TPO −1/2, 2 Z 9/2. 4F UJFOF FOUPODFT RVF FM QPMJOPNJP RVFEB GBDUPSJ[BEP DPNP (x + 1/2)(x − 2)(x − 9/2). &KFSDJDJP " )BMMF MPT DFSPT SBDJPOBMFT TJ FYJTUFO EFM QPMJOPNJP p(x) = x6 − x5 − 8x4 − 2x3 + 17x2 + 19x + 6.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFNQMP " %FNPTUSFNPT RVF OP FYJTUF VO OÙNFSP SBDJPOBM x UBM RVF x2 = 3/5. 3B[POFNPT QPS DPOUSBEJDDJÓO Z TVQPOHBNPT RVF TÎ FYJTUF &O UPODFT EJDIP OÙNFSP FT VO DFSP SBDJPOBM EFM QPMJOPNJP p(x) = 5x2 − 3. 1FSP MPT ÙOJDPT DFSPT SBDJPOBMFT EF FTUF QPMJOPNJP TPO ±1, ±3, ±1/5 Z ±3/5 6O DIFRVFP EJSFDUP NVFTUSB RVF OJOHVOP EF FMMPT BOVMB FM QPMJ OPNJP Z QPS UBOUP OP FYJTUF VO SBDJPOBM DPO FTB QSPQJFEBE √ √ &KFSDJDJP " 1SVFCF RVF 2 + 3 FT VO OÙNFSP JSSBDJPOBM .VZ JNQPSUBOUF FM SFTVMUBEP RVF QSFTFOUBNPT B DPOUJOVBDJÓO DVZB EFNPTUSBDJÓO QVFEF TFS DPOTVMUBEB FO MB SFGFSFODJB QÃH  5FPSFNB " 5FPSFNB GVOEBNFOUBM EFM ÃMHFCSB  4FB p(x) VO QPMJ OPNJP FO C[x] EF HSBEP n ≥ 1 &OUPODFT p(x) UJFOF VOB SBÎ[ FO MPT DPNQMFKPT  $PSPMBSJP " 5PEP QPMJOPNJP FO C[x] EF HSBEP n ≥ 1 UJFOF FYBDUB NFOUF n SBÎDFT DPNQMFKBT %FNPTUSBDJÓO 4FB p(x) = an xn + an−1 xn−1 + · · · + a0 ∈ C[x] 1PS FM UFPSFNB GVOEBNFOUBM EFM ÃMHFCSB FYJTUF VOB SBÎ[ r1 ∈ C Z QPS FM UFPSFNB EFM GBDUPS FYJTUF VO ÙOJDP QPMJOPNJP c1 (x) EF HSBEP n−1 EF NBOFSB RVF p(x) = c1 (x)(x−r1 ). "QMJDBOEP OVFWBNFOUF FM UFPSFNB GVOEBNFOUBM EFM ÃMHFCSB BM QPMJOPNJP c1 (x) FODPOUSBNPT r2 ∈ C Z VO ÙOJDP QPMJOPNJP c2 (x) ∈ C[x] EF HSBEP n − 2 UBM RVF p(x) = c2 (x)(x − r1 )(x − r2 ). &M SB[POBNJFOUP TF SFQJUF FO VO OÙNFSP àOJUP EF QBTPT ZB RVF ;`/ (p) = n IBTUB DPODMVJS RVF p(x) = an (x − r1 )(x − r2 ) · · · (x − rn ). 'JOBMNFOUF OP QVFEFO FYJTUJS NÃT EF n SBÎDFT ZB RVF TJ α ∈ C DPO  α = ri QBSB UPEP i = 1, 2, . . . , n FOUPODFT p(α) = 0.  %F MB QSVFCB BOUFSJPS OP RVFEB DMBSP TJ MBT SBÎDFT TPO EJGFSFOUFT EF IFDIP QVFEFO TFS UPEBT JHVBMFT DPNP MP FWJEFODJBO NVDIPT FKFNQMPT p(x) = x3 − 3x2 + 3x − 1 = (x − 1)3 UJFOF DPNP SBÎDFT r1 = r2 = r3 = 1 &TUB TJUVBDJÓO NPUJWB FM TJHVJFOUF DPODFQUP %FàOJDJÓO " 4FBO K VO DBNQP Z p ∈ K[x] VO QPMJOPNJP OP DPOT UBOUF %FDJNPT RVF VO FMFNFOUP a ∈ K FT VO DFSP EF NVMUJQMJDJEBE m ≥ 1 TJ (x − a)m | p(x) Z (x − a)m+1  p(x) 4J m = 1 TF BDPTUVNCSB EFDJS RVF a FT VOB SBÎ[ TJNQMF





&M UFPSFNB GVOEBNFOUBM EFM ÃMHFCSB GVF EFNPTUSBEP QPS $BSM 'SJFESJDI (BVTT   B MB FEBE EF WFJOUJEÓT BÒPT (BVTT IJ[P BQPSUFT TJHOJàDBUJWPT FO WBSJBT EJTDJ QMJOBT UFPSÎB EF OÙNFSPT HFPNFUSÎB BOÃMJTJT BTUSPOPNÎB Z GÎTJDB 5PEPT TVT USBCBKPT EBO QBSB EPDF WPMÙNFOFT

"QÊOEJDF

5FOJFOEP FO DVFOUB FTUB EFàOJDJÓO FOUSF MBT n SBÎDFT EF VO QPMJOP NJP EF HSBEP n TF DPOUFNQMB MB NVMUJQMJDJEBE EF DBEB SBÎ[ Z BTÎ EJDIP QPMJOPNJP QVFEF FTDSJCJSTF FO MB GPSNB an (x − r1 )n1 (x − r2 )n2 · · · (x − rk )nk , EPOEF n1 + n2 + · · · + nk = n. &M TJHVJFOUF UFPSFNB NVFTUSB RVF MBT SBÎDFT DPNQMFKBT EF VO QPMJOP NJP DPO DPFàDJFOUFT SFBMFT WJFOFO QPS QBSFT 5FPSFNB " 4J a + ib FT VO DFSP EF VO QPMJOPNJP p(x) ∈ R[x] FOUPODFT TV DPOKVHBEP a − ib UBNCJÊO FT VO DFSP EF EJDIP QPMJOPNJP %FNPTUSBDJÓO &T DMBSP RVF TJ b = 0 FM UFPSFNB FT JONFEJBUP QPS UBOUP QPEFNPT TVQPOFS RVF b = 0. 1SPCFNPT RVF (x − a − ib)(x − a + ib) = (x − a)2 + b2 FT GBDUPS EF p(x) EF MP DVBM TF DPODMVZF RVF p(a − ib) = 0 1PS FM BMHPSJUNP EF MB EJWJTJÓO FYJTUFO ÙOJDPT QPMJOPNJPT FM DPDJFOUF q(x) Z FM SFTJEVP EF MB GPSNB rx + R UBMFT RVF p(x) = q(x)[(x − a)2 + b2 ] + rx + R. $PNP p(a + ib) = 0 FOUPODFT r(a + ib) + R = 0 Z QPS UBOUP ra + R = 0 = rb. $PNP b = 0 FOUPODFT r = 0 = R MP DVBM QSVFCB RVF   p(x) = q(x)[(x − a)2 + b2 ]. 0CTFSWBDJÓO &M ÙMUJNP UFPSFNB JNQMJDB RVF UPEP QPMJOPNJP EF HSBEP JNQBS DPO DPFàDJFOUFT FO R UJFOF QPS MP NFOPT VOB SBÎ[ SFBM √ &KFNQMP " 4J 12 (1 + i 3) FT VO DFSP EFM QPMJOPNJP p(x) = 3x4 − 10x3 + 4x2 − x − 6, FODPOUSFNPT MBT PUSBT SBÎDFT 1PS FM UFPSFNB QSFWJP UFOFNPT RVF √ 1 (1 − i 3) UBNCJÊO FT SBÎ[ EF p(x). 1BSB IBMMBS MBT PUSBT EPT SBÎDFT 2 CBTUB FODPOUSBS MPT DFSPT EF q(x) EPOEF √ √ p(x) = q(x)(x − 1/2 − i 3/2)(x − 1/2 + i 3/2) = q(x)(x2 − x + 1). 1PS EJWJTJÓO PSEJOBSJB EF QPMJOPNJPT FODPOUSBNPT RVF q(x) = 3x2 − 7x − 6, DVZBT TPMVDJPOFT TPO 3 Z −2/3 1PS UBOUP MPT DFSPT EF p(x) TPO √ 1 (1 ± i 3), 3 Z −2/3 &T JNQPSUBOUF BOPUBS RVF UBNCJÊO QVEP VTBS 2 TF FM 5FPSFNB " QBSB IBMMBS FTUPT ÙMUJNPT DFSPT



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

&KFSDJDJP " &ODVFOUSF VO QPMJOPNJP EF HSBEP 3 Z DPO DPFàDJFOUFT SFBMFT UBM RVF 1 − i FT VOP EF TVT DFSPT y&T ÙOJDP EJDIP QPMJOPNJP 6O IFDIP NVZ QBSFDJEP BM 5FPSFNB " FT QSFTFOUBEP B DPOUJOVB DJÓO 4V EFNPTUSBDJÓO VTB JEFBT TJNJMBSFT B MB QSVFCB EF EJDIP UFPSFNB Z FT EFKBEB DPNP FKFSDJDJP 5FPSFNB " 4FBO p(x) ∈ Q[x] DPO DPFàDJFOUFT SBDJP √ VO QPMJOPNJP √ / Q 4J a + b FT VO DFSP EF p(x) OBMFT Z a, b ∈√Q UBMFT RVF a + b ∈ FOUPODFT a − b UBNCJÊO FT VO DFSP EF EJDIP QPMJOPNJP -B DPOEJDJÓO SFTQFDUP B MPT DPFàDJFOUFT FO FM UFPSFNB BOUFSJPS OP TF QVFEF FMJNJOBS DPNP TF FWJEFODJB FO MB TJHVJFOUF JMVTUSBDJÓO TFB √ p(x) = 2 x2 − 3x − 1. √ 2) = 0 QFSP $ÃMDVMPT EJSFDUPT Z TJNQMFT NVFTUSBO RVF p(1 + √ p(1 − 2) = 0. &KFSDJDJP " 4FBO a1 , a2 , . . . , an OÙNFSPT DPNQMFKPT $ÃMDVMPT EJSFDUPT BSSPKBO RVF (x + a1 )(x + a2 ) = x2 + (a1 + a2 ) x + a1 a2 (x + a1 )(x + a2 )(x + a3 ) = x3 + (a1 + a2 + a3 ) x2 + (a1 a2 + a1 a3 + a2 a3 ) x + a1 a2 a3 (x + a1 )(x + a2 )(x + a3 )(x + a4 ) = x4 + (a1 + a2 + a3 + a4 ) x3 + (a1 a2 + a1 a3 + a1 a4 + a2 a3 + a2 a4 + a3 a4 ) x2 + (a1 a2 a3 + a1 a2 a4 + a2 a3 a4 + a1 a3 a4 ) x + a 1 a 2 a 3 a 4 . y%FTDVCSJÓ FM QBUSÓO /PUF RVF MB ÙMUJNB JHVBMEBE FT FRVJWBMFOUF B MB JHVBMEBE (x + a1 )(x + a2 )(x + a3 )(x + a4 ) = x4 + S1 x3 + S2 x2 + S3 x + S4 , EPOEF S1 FT MB TVNB EF MPT OÙNFSPT a1 , a2 , a3 , a4 ; S2 FT MB TVNB EF MPT QSPEVDUPT EF FTUPT DVBUSP OÙNFSPT UPNBEPT EF EPT FO EPT S3 FT MB TVNB EF MPT QSPEVDUPT EF FTUPT DVBUSP OÙNFSPT UPNBEPT EF USFT FO USFT Z S4 FT MB TVNB EF MPT QSPEVDUPT EF FTUPT DVBUSP OÙNFSPT UPNBEPT UPEPT FTUB TVNB DPOUJFOF VO TPMP UÊSNJOP QVFT 4 C4 = 1  -B JEFB EF FTUF FKFSDJDJP FT EFNPTUSBS QPS JOEVDDJÓO QPS FKFNQMP RVF (x + a1 )(x + a2 ) · · · (x + an ) = xn + S1 xn−1 + S2 xn−2 + S3 xn−3 + · · · + Sn−1 x + Sn , "



"QÊOEJDF

EPOEF Sr EFOPUB MB TVNB EF MPT QSPEVDUPT EF MPT n OÙNFSPT a1 , . . . , an UPNBEPT EF r FO r, DPO 1 ≤ r ≤ n. " DPOUJOVBDJÓO WBNPT B VTBS FM FKFSDJDJP QSFWJP QBSB EFEVDJS VO JNQPSUBOUF IFDIP RVF QFSNJUF EFTBSSPMMBS VOB QPUFODJB OBUVSBM EF VO CJOPNJP 1BSB FTUP OPUFNPT JOJDJBMNFOUF RVF MB TVNB Sr FTUÃ DPNQVFTUB QPS n Cr UÊSNJOPT Z QPS UBOUP TJ IBDFNPT a1 = a2 = · · · = an = a FOUPODFT Sr = n Cr ar DPO MP DVBM QPEFNPT DPODMVJS RVF (x + a)n = xn + n C1 axn−1 + n C2 a2 xn−2 + n C3 a3 xn−3 + · · · + n Cn−1 an−1 x + an n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3 x x = xn + nxn−1 a + a + a 2! 3! + · · · + an n    n n−k k x = a . "

k k=0

&TUB GÓSNVMB SFDJCF SFDJCF FM OPNCSF EF UFPSFNB EFM CJOPNJP n 0CTFSWF n NPT RVF DPO k = 0 TF UJFOF FM QSJNFS UÊSNJOP x QVFT   0 = 1 Z DPO k = n TF UJFOF FM (n + 1)ÊTJNP UÊSNJOP an ZB RVF nn = 1. &KFSDJDJP " %FNVFTUSF MBT TJHVJFOUFT JHVBMEBEFT   n   n   n n k n =2 = 0. Z (−1) k k k=0

k=0

&O FM UFPSFNB EFM CJOPNJP DBNCJBNPT a QPS −a Z PCUFOFNPT FM NJT NP EFTBSSPMMP QFSP DPO MPT TJHOPT BMUFSOBEPT FNQF[BOEP DPO UÊSNJOP xn QSFDFEJEP EF TJHOP + Z UFSNJOBOEP DPO an QSFDFEJEP EF (−1)n  .ÃT QSFDJTBNFOUF     n n−1 n n−2 2 n n x a+ x a + · · · + (−1)n an . (x − a) = x − 1 2 $PO FTUBT JEFBT FO NFOUF WBNPT B PDVQBSOPT BIPSB EF NPTUSBS MB SF MBDJÓO FOUSF MPT DPFàDJFOUFT Z MBT SBÎDFT EF VO QPMJOPNJP EF HSBEP n &TUB SFMBDJÓO FT EF NVDIB BZVEB DVBOEP TF DPOPDF EF NBOFSB QSFWJB JOGPSNBDJÓO EF MBT SBÎDFT EF VO QPMJOPNJP 4FB p(x) VO QPMJOPNJP FM DVBM TJO QÊSEJEB EF HFOFSBMJEBE QPEFNPT TVQPOFS RVF FT EF MB GPSNB p(x) = xn + An−1 xn−1 + An−2 xn−2 + · · · + A1 x + A0 . 4FBO r1 , r2 , · · · , rn MPT DFSPT EF FTUF QPMJOPNJP &OUPODFT (x − r1 )(x − r2 ) · · · (x − rn ) = 0.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

6TBOEP MB SFMBDJÓO " Z MB JHVBMEBE FOUSF QPMJOPNJPT DPODMVJNPT RVF −A1 = r1 + r2 + · · · + rn A2 = -B TVNB EF MPT QSPEVDUPT EF MBT n SBÎDFT UPNBEBT EF EPT FO EPT −A3 = -B TVNB EF MPT QSPEVDUPT EF MBT n SBÎDFT UPNBEBT EF USFT FO USFT   (−1)n A0 = r1 r2 · · · rn . &KFNQMP " 3FTPMWBNPT MB FDVBDJÓO 4x3 − 24x2 + 23x + 18 = 0 TBCJFOEP RVF MBT SBÎDFT FTUÃO FO QSPHSFTJÓO BSJUNÊUJDB %FOPUFNPT QPS a − d, a Z a + d MBT SBÎDFT EF MB FDVBDJÓO FRVJWBMFOUF x3 − 6x2 +

9 23 x + = 0. 4 2

1PS MB SFMBDJÓO FOUSF MPT DPFàDJFOUFT Z MPT DFSPT EF VO QPMJOPNJP FODPO USBNPT RVF 3a = 6 2

a(a − d) + a(a + d) + (a − d2 ) = 23/4 a(a2 − d2 ) = −9/2. %F MB QSJNFSB Z DVBMRVJFSB EF MBT PUSBT EPT FDVBDJPOFT TF PCUJFOF RVF a = 2 Z d2 = 25/4 "OBMJ[BOEP MBT EPT QPTJCJMJEBEFT QBSB d MMFHBNPT B RVF MBT SBÎDFT CVTDBEBT TPO −1/2, 2 Z 9/2. &KFSDJDJP " &ODVFOUSF MPT DFSPT EFM QPMJOPNJP 18x3 + 81x2 + 121x + 60, TBCJFOEP RVF VOB EF MBT SBÎDFT FT MB TFNJTVNB EF MBT PUSBT EPT &KFSDJDJP " 4FBO a, b Z c MBT SBÎDFT DPNQMFKBT EF 2x3 − 3x2 + 4x + 1. )BMMBS J a + b + c,

ab + ac + bc Z abc.

JJ a2 + b2 + c2 ,

a 3 + b3 + c 3 Z a 4 + b 4 + c 4 .

JJJ a2 b2 + a2 c2 + b2 c2 ,

1 1 1 1 1 1 + + Z 2 + 2 + 2. a b c a b c

&KFSDJDJP " )BMMF MB TVNB EF MBT DVBSUBT QPUFODJBT EF MBT SBÎDFT EF x3 + qx + r FO UÊSNJOPT EF q P r 



"QÊOEJDF

&KFSDJDJP " )BMMBS MB DPOEJDJÓO QBSB RVF FM QPMJOPNJP x3 − px2 + qx − r UFOHB TVT SBÎDFT FO QSPHSFTJÓO HFPNÊUSJDB &KFSDJDJP " JBMBT`Qv2+iQ R  "WFSJHVF TPCSF MB SFHMB EF MPT TJHOPT EF %FTDBSUFT Z BQMÎRVFMB QBSB EFUFSNJOBS FM OÙNFSP EF QPTJCMFT SBÎDFT QPTJUJWBT OFHBUJWBT Z DPNQMFKBT EF MPT TJHVJFOUFT QPMJOPNJPT J 2x6 + 3x4 + 2x2 + 9 JJ x5 − 2x3 + 5x − 7 JJJ 3x4 + 12x2 + 5x − 4 &KFSDJDJP " 1SVFCF RVF FM QPMJOPNJP 2x7 − x4 + 4x3 − 5 UJFOF BM NFOPT  SBÎDFT DPNQMFKBT &KFSDJDJP " JBMBT`Qv2+iQ k  *OEBHVF TPCSF MB EFTDPNQPTJDJÓO FO GSBDDJPOFT QBSDJBMFT P TJNQMFT Z BQMÎRVFMB FO MBT TJHVJFOUFT GSBDDJPOFT QBSB IBMMBS TV EFTDPNQPTJDJÓO FO TVNB EF GSBDDJPOFT TJNQMFT SFBMFT x3 − 6x2 + 10x − 5 x4 − 7x3 + 17x2 − 21x + 18

5x − 11 2x2 + x − 6 3x2 + x − 2 (x − 2)2 (1 − 2x)

4x2 − 5x − 15 x3 − 4x2 − 5x

42 − 19x 2 (x + 1)(x − 4)

x2 + x − 1 (x2 + 1)2 (x − 2)

6x3 + 5x2 − 7 3x2 − 2x − 1

2x4 − 2x3 + 6x2 − 5x + 1 . x3 − x2 + x − 1

" 3FQSFTFOUBDJÓO EFDJNBM EF VO OÙNFSP SFBM &O FTUB TFDDJÓO OPT WBNPT B PDVQBS EFM EFTBSSPMMP EFDJNBM Z FO PUSBT CBTFT EF VO OÙNFSP SFBM QSJNFSP FO FM DBTP EF VO FOUFSP N Z MVFHP QBSB DVBMRVJFS SFBM x 4JO QFSEFS HFOFSBMJEBE TVQPOHBNPT RVF N > 0 1PS FM BMHPSJUNP EF MB EJWJTJÓO FYJTUFO ÙOJDPT FOUFSPT q ≥ 0 Z 0 ≤ r < 10 UBMFT RVF N = 10 · q + r. 4J q < 10 FM EFTBSSPMMP EFDJNBM EFM OBUVSBM N FTUÃ EBEP QPS N = 10 · q + r · 100 F JEFOUJàDBNPT 10 · q + r · 100 DPO MB FTDSJUVSB q r OP FT FM QSPEVDUP FOUSF q Z r Z BTÎ N = q r = 10 · q + r · 100 .



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-MBNBNPT BM OBUVSBM q FM EÎHJUP EF MBT EFDFOBT Z BM OBUVSBM r MP MMBNBNPT FM EÎHJUP EF MBT VOJEBEFT "IPSB CJFO TJ q > 10 FOUPODFT QPEFNPT BQMJDBS OVFWBNFOUF FM BMHPSJUNP EF MB EJWJTJÓO QBSB PCUFOFS q = 10 · q1 + r1 QBSB ÙOJDPT OBUVSBMFT q1 Z 0 ≤ r1 < 10. 3FFNQMB[BOEP FTUP MMFHBNPT B N = 102 · q1 + 10 · r1 + r · 100 .

"

4J q1 < 10 FOUPODFT " FT MB SFQSFTFOUBDJÓO EFDJNBM QBSB N Z MMBNB NPT B q1 FM EÎHJUP EF MBT DFOUFOBT B r1 FM EÎHJUP EF MBT EFDFOBT Z B r FM EÎHJUP EF MBT VOJEBEFT $PNP BOUFT JEFOUJàDBNPT MB JHVBMEBE FO "

DPO MB FTDSJUVSB N = q1 r1 r. 4J QPS FM DPOUSBSJP q1 > 10 FM QSPDFTP TF SFQJUF VO OÙNFSP àOJUP EF WFDFT IBTUB DPOTFHVJS VOB FYQSFTJÓO EF MB GPSNB N = 10j · qj−1 + 10j−1 · rj−1 + · · · + 10 · r1 + r · 100 .

"

-B FYQSFTJÓO " FT MB SFQSFTFOUBDJÓO EFDJNBM QBSB N Z TF JEFOUJàDB FTUB JHVBMEBE DPO MB FTDSJUVSB N = qj−1 rj−1 · · · r1 r 1PS FKFNQMP 7289 = 7 · 103 + 2 · 102 + 8 · 101 + 9 · 100 . 'JKFNPT BIPSB VO OÙNFSP SFBM x 7FBNPT RVF FYJTUF VOB ÙOJDB SFQSF TFOUBDJÓO EFDJNBM EF x FO VO TFOUJEP RVF QSFDJTBSFNPT NÃT BEFMBOUF #BTUB TVQPOFS RVF 1 > x ≥ 0 QVFT QBSB x < 0 UFOJFOEP FO DVFOUB RVF x = [[ x ]] + ((x)) EPOEF ((x)) ∈ [0, 1) FT MB QBSUF GSBDDJPOBSJB EF x IBMMBNPT MB SFQSFTFOUBDJÓO EFDJNBM EF ((x)) Z DPO MP FYQVFTUP QSF WJBNFOUF QBSB [[ x ]] TF PCUJFOF MB FYQBOTJÓO EFDJNBM DPSSFTQPOEJFOUF B x < 0. 4FB n1 FM NBZPS FOUFSP OP OFHBUJWP UBM RVF n1 /10 ≤ x 5BM FOUFSP FT n1 = [[ 10 x ]] RVF FWJEFOUFNFOUF BSSPKB MBT EFTJHVBMEBEFT n1 /10 ≤ x < (n1 + 1)/10. $PNP x ∈ [0, 1) FOUPODFT 0 ≤ 10 x < 10 Z QPS UBOUP 0 ≤ n1 < 10 FTUP FT 0 ≤ n1 ≤ 9. 4FB r1 := n1 /10 &T DMBSP RVF 0 ≤ r1 Z QPS MBT EFTJHVBMEBEFT QSFWJBT TF UJFOF RVF "

r1 ≤ x < r1 + 1/10. 4FB n2 FM NBZPS FOUFSP OP OFHBUJWP UBM RVF r1 + n2 /102 ≤ x 5BM FOUFSP FTUÃ EBEP QPS n2 = [[ 102 (x − r1 ) ]] RVF TJHOJàDB MBT EFTJHVBMEBEFT r1 + n2 /102 ≤ x < r1 + (n2 + 1)/102 .



"QÊOEJDF

%FàOJNPT r2 := r1 + n2 /102 = n1 /10 + n2 /102  &T DMBSP RVF r1 ≤ r2 Z QPS MBT EFTJHVBMEBEFT QSFWJBT TF UJFOF RVF r2 ≤ x < r2 + 1/102 . %F " TF UJFOF RVF 0 ≤ x − r1 < 1/10 Z QPS UBOUP 0 ≤ 102 (x − r1 ) < 10 Z BTÎ 0 ≤ n2 ≤ 9 1SPDFEFNPT QPS JOEVDDJÓO TVQPOHBNPT RVF IFNPT FTDPHJEP n1 , n2 , . . . , nk DPO 0 ≤ nj ≤ 9 QBSB j = 1, 2, . . . , k Z UBNCJÊO RVF IFNPT FODPOUSBEP 0 := r0 ≤ r1 ≤ r2 ≤ · · · ≤ rk UBMFT RVF rk = rk−1 + nk /10k

Z

nk = [[ 10k (x − rk−1 ) ]].

%F MB FYQSFTJÓO QBSB nk TF EFTQSFOEF RVF rk−1 + nk /10k ≤ x < rk−1 + (nk + 1)/10k , MP DVBM JNQMJDB RVF rk ≤ x < rk + 1/10k .

"

%FàOBNPT nk+1 := [[ 10k+1 (x − rk ) ]] &OUPODFT nk+1 ≤ 10k+1 (x − rk ) < nk+1 + 1, FT EFDJS rk + nk+1 /10k+1 ≤ x < rk + (nk+1 + 1)/10k+1 . -VFHP TJ EFàOJNPT rk+1 := rk + nk+1 /10k+1 FOUPODFT rk+1 ≤ x < rk+1 + 1/10k+1 . /PUFNPT RVF " JNQMJDB RVF nk+1 ≥ 0 Z UBNCJÊO TF EFTQSFOEF RVF 0 ≤ nk+1 ≤ 9. &O DPOTFDVFODJB rk ≤ rk+1 . %F FTUB NBOFSB IFNPT DPOTUSVJEP TVDFTJPOFT {nk } ⊂ N Z {rk } ⊂ R UBMFT RVF 0 ≤ nk ≤ 9, rk ≤ rk+1 Z BEFNÃT rk = n1 /10 + n2 /102 + · · · + nk /10k ,

rk ≤ x < rk + 1/10k

Z nk = [[ 10k (x − rk−1 ) ]]. %F FTUP TF UJFOF FOUPODFT RVF rk −→ x DVBOEP k → ∞. 3FTVNJNPT MP BOUFSJPS FO FM TJHVJFOUF UFPSFNB



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

5FPSFNB " 4FB 0 ≤ x < 1 &YJTUF VOB TVDFTJÓO EF OBUVSBMFT {nk } UBM RVF QBSB DBEB OBUVSBM k ≥ 1, 0 ≤ nk ≤ 9 &YJTUF UBNCJÊO VOB TVDFTJÓO EF SFBMFT {rk } UBM RVF r0 = 0 Z QBSB DBEB OBUVSBM k ≥ 1 rk = n1 /10 + n2 /102 + · · · + nk /10k . "EFNÃT nk = [[ 10k (x − rk−1 ) ]] Z rk −→ x DVBOEP k → ∞. 4F BDPTUVNCSB VTBS MB FTDSJUVSB x = 0, n1 n2 · · · nk · · · Z MB MMBNBNPT SFQSFTFOUBDJÓO EFDJNBM EFM SFBM x ∈ [0, 1) &TUB FTDSJUVSB TF JEFOUJàDB DPO n1 /10 + n2 /102 + · · · + nk /10k + · · · . &O FM DBTP HFOFSBM QBSB x ≥ 0 EBEP VOB FYQBOTJÓO EFDJNBM FTUÃ EBEB QPS x = [[ x ]], n1 n2 · · · nk · · · = [[ x ]] + n1 /10 + n2 /102 + · · · + nk /10k + · · · . &KFNQMP " y2VÊ OÙNFSP SFBM EFOPUB MB SFQSFTFOUBDJÓO EFDJNBM 0, 111111 · · · %FOPUFNPT EJDIP OÙNFSP QPS x /PUFNPT RVF nk = 1 QBSB UPEP k ≥ 1 -VFHP x = MÎN rk = MÎN (1/10 + 1/102 + · · · + 1/10k ) k→∞

k→∞

= (1/10) MÎN (1 + 1/10 + 1/102 + · · · + 1/10k−1 ) k→∞

1 − 1/10k = (1/9) MÎN (1 − 1/10k ) k→∞ k→∞ 9/10

= (1/10) MÎN = 1/9.

&KFNQMP " )BMMFNPT MB SFQSFTFOUBDJÓO EFDJNBM EF x = −17/9. 0CTFSWFNPT RVF x = −17/9 = [[ x ]] + ((x)) = −2 + 1/9 Z QPS UBOUP CBTUB IBMMBS MB SFQSFTFOUBDJÓO EFDJNBM EF MB QBSUF GSBDDJPOBSJB ((x)) = 1/9 RVF QPS FM FKFNQMP QSFWJP FT 0, 11111111 · · · . &O DPODMVTJÓO MB SFQSFTFOUBDJÓO EFDJNBM EF −17/9 FT [[ x ]] + 0, 111111 · · · = −1, 8888888 · · · 0CTFSWBDJPOFT  4FBO x ∈ [0, 1) Z k ∈ N &OUPODFT J [[ 10k x ]] = 10k rk . &TUP FT DPOTFDVFODJB EF MBT EFTJHVBMEBEFT 10k rk ≤ 10k x < 10k rk + 1 RVF TF PCUJFOFO EF "

Z EF MB JHVBMEBE 10k rk = 10k (n1 /10 + n2 /102 + · · · + nk /10k ) ∈ N.



"QÊOEJDF

JJ %FM MJUFSBM QSFWJP TF EFTQSFOEF PUSB GPSNB EF PCUFOFS nk  .ÃT QSF DJTBNFOUF [[ 10k x ]] − 10[[ 10k−1 x ]] = nk QBSB UPEP OBUVSBM k ≥ 1, ZB RVF [[ 10k x ]] − 10[[ 10k−1 x ]] = 10k rk − 10 · 10k−1 rk−1 = 10k (rk − rk−1 ) = nk .  %BEP VO OÙNFSP SFBM x ≥ 0 1PS MB NBOFSB DPNP IFNPT PCUFOJEP VO EFTBSSPMMP EFDJNBM TF UJFOF RVF EJDIP OÙNFSP OP QPTFF SFQSFTFOUB DJÓO EFDJNBM DPO VOB DPMB GPSNBEB TPMBNFOUF QPS FM EÎHJUP  &O FGFDUP TVQPOHBNPT RVF FYJTUJFSBO x ∈ R Z p ∈ N∗ UBMFT RVF VO EFTBSSPMMP EFDJNBM EF x GVFTF EF MB GPSNB x = n0 , n1 n2 · · · np 99999 · · · = n0 + n1 /10 + n2 /102 + · · · + np /10p + 9/10p+1 + 9/10p+2 + · · · , DPO n0 = [[ x ]]. &M TJHVJFOUF BSHVNFOUP NPTUSBSÎB RVF 10p x ∈ N QBSB UPEP m > p n0 + rm = n0 + n1 /10 + n2 /102 + · · · + np /10p + 9/10p+1 + 9/10p+2 + · · · + 9/10m =

p  ni + 9/10p+1 (1 + 1/10 + 1/102 + · · · + 1/10m−p−1 ) 10i i=0

p m−p  ni p+1 1 − (1/10) = + 9/10 10i 9/10 i=0

p  ni = + 1/10p − 1/10m . 10i i=0

)BDJFOEP RVF m → ∞ DPODMVJSÎBNPT RVF x =

p  ni + 1/10p Z BTÎ 10i i=0

10p x = 10p (n0 + n1 /10 + n2 /102 + · · · + np /10p ) + 1 ∈ N. 6TBOEP FTUP Z MB QBSUF JJ EFM OVNFSBM BOUFSJPS MMFHBSÎBNPT B MB DPOUSB EJDDJÓO 9 = np+2 = [[ 10p+2 x ]] − 10[[ 10p+1 x ]] = 10p+2 x − 10p+2 x = 0.



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

-VFHP OP FYJTUF x ∈ R DVZB SFQSFTFOUBDJÓO EFDJNBM UFOHB DPMB GPSNBEB QPS FM EÎHJUP   1PS MP BOUFSJPS Z DPNP [[ 10k x ]] − 10[[ 10k−1 x ]] = nk CBKP FTUB DPOTUSVDDJÓO FM EFTBSSPMMP EFDJNBM EF VO OÙNFSP SFBM x EBEP FT ÙOJDP TJ OP BENJUJNPT BRVFMMPT RVF UJFOFO DPMB GPSNBEB QPS FM EÎHJUP  4JO FNCBSHP FT QPTJCMF FODPOUSBS EPT FYQBOTJPOFT EFDJNBMFT EJGFSFOUFT RVF EFOPUFO FM NJTNP SFBM x 1PS FKFNQMP 0, 500000 · · · = 0, 499999 · · · =

1 · 2

 0CTFSWFNPT àOBMNFOUF RVF FO MB DPOTUSVDDJÓO EFM EFTBSSPMMP EF DJNBM EF VO OÙNFSP SFBM x FM EÎHJUP 10 QVFEF TFS TVTUJUVJEP QPS DVBMRVJFS OBUVSBM b > 1 4J b > 10 TF VTBO MFUSBT NBZÙTDVMBT A, B, C, D, E, F FO MVHBS EF b. %FDJNPT FOUPODFT RVF x = n0 + n1 /b + n2 /b2 + n3 /b3 + · · · FT MB SFQSFTFOUBDJÓO FO CBTF b EFM OÙNFSP SFBM x $PNP TF NFODJPOÓ DPO FTUB DPOTUSVDDJÓO OP TF QSFTFOUB FM DBTP EF OÙNFSPT DPO EFTBSSPMMP RVF BENJUBO DPMB GPSNBEB QPS FM EÎHJUP b − 1 6TBNPT MB OPUBDJÓO (x)b = n0 + n1 /b + n2 /b2 + n3 /b3 + · · ·

Z

(x)10 ≡ x.

&KFNQMP " "M BQMJDBS FM BMHPSJUNP EF MB EJWJTJÓO FODPOUSBNPT RVF (15)2 = 1111. /PUFNPT RVF 1 · 23 + 1 · 22 + 1 · 21 + 1 · 20 = 15. &KFNQMP " )BMMFNPT FM EFTBSSPMMP UFSOBSJP EF 1/2 FT EFDJS (1/2)3  &O FTUF DBTP nk ∈ {0, 1, 2}, n0 = [[ 1/2 ]] = 0 Z QBSB k ≥ 1, nk = [[ 3k (1/2 − rk−1 ) ]]

Z

rk = n1 /3 + n2 /32 + · · · + nk /3k .

-VFHP n1 = 1 Z r1 = 1/3 JNQMJDBO RVF n2 = [[ 32 (1/2 − 1/3) ]] = 1 7FBNPT QPS JOEVDDJÓO RVF nk = 1 QBSB UPEP k ≥ 1. 4VQPOHBNPT RVF n1 = n2 = · · · = nk = 1 Z WFBNPT RVF nk+1 = 1. &O FGFDUP   1 1 − 1/3k 2 k = 1/2 − 1/(2 · 3k ) rk = 1/3 + 1/3 + · · · + 1/3 = 3 2/3 HBSBOUJ[B RVF rk+1 = [[ 3k+1 (1/2 − rk ) ]] = [[ 3/2 ]] = 1 &O DPOTFDVFODJB (1/2)3 = 0, 111111 · · · &KFSDJDJP " &ODVFOUSF MBT TJHVJFOUFT SFQSFTFOUBDJPOFT 1/8, (2/5)2 . &KFSDJDJP " &ODVFOUSF FM OÙNFSP SFBM DPSSFTQPOEJFOUF B (0, 122222 · · · )4 .



±OEJDF BMGBCÊUJDP

A BDPUBEP  JOGFSJPSNFOUF  TVQFSJPSNFOUF  BMHPSJUNP EF &VDMJEFT  BMHPSJUNP EF MB EJWJTJÓO   BSHVNFOUP QSJODJQBM  BSRVJNFEJBOB   BTPDJBUJWB  BYJPNB EF DBNQP  EF DFSSBEVSB  EF DPNQMFUF[  EF DPOUJOVJEBE  EF FMFDDJÓO  EF PSEFO  EF USJDPUPNÎB  BYJPNB EF DPNQMFUF[  BYJPNBT EF 1FBOP 

B CBTF C  CPMB 

C DBNQP  BSRVJNFEJBOP  DBSBDUFSÎTUJDB EF 

DPNQMFUP  PSEFOBEP  TVCDBNQP  DBSEJOBM  DFSP EF VO QPMJOPNJP  DMBTF EF FRVJWBMFODJB   DMBTFT SFTJEVBMFT  DMBVTVSB  DMBVTVSBUJWB  DPMB EF VOB TVDFTJÓO  DPNCJOBDJÓO   DPNQBDUP  DPOHSVFODJB  DPOKVHBEP  DPOKVOUP JOEVDUJWP  DPOKVOUP BCJFSUP  DPOKVOUP DFSSBEP  DPOKVOUP EF $BOUPS  DPOKVOUP EF PQVFTUPT  DPOKVOUP àOJUP  DPOKVOUP JOàOJUP  DPONVUBUJWB  DPOUJOVJEBE EFM NÓEVMP  DPQSJNPT  DPSUBEVSB  DPUB JOGFSJPS  DPUB TVQFSJPS  DVBESBEP QFSGFDUP 



5ÓQJDPT QSFWJPT B MB NBUFNÃUJDB TVQFSJPS

DVCSJNJFOUP BCJFSUP  DVFSQP 

H

D

I

EÎHJUP EF MBT DFOUFOBT  EÎHJUP EF MBT EFDFOBT  EÎHJUP EF MBT VOJEBEFT  EFTJHVBMEBE EF #FSOPVMMJ   EFTJHVBMEBE USJBOHVMBS    EJBHSBNB EF "SHBOE  EJGFSFODJB  EJWJTPS  EJWJTPSFT EF DFSP 

E FKF JNBHJOBSJP  FKF SFBM  FMFNFOUP OFVUSP  FOUFSP JNQBS  FOUFSP QBS  FOUPSOP  FRVJQPUFOUFT 

F GÓSNVMB %F .PJWSF  GBDUPSJBM  GSPOUFSB  GVODJÓO BDPUBEB  GVODJÓO EF %JSJDIMFU  GVODJÓO FYQPOFODJBM  GVODJÓO 4VDFTPS 

G HSBEP EF VO QPMJOPNJP  HSVQP  BCFMJBOP  HSVQPJEF 



IPNPNPSàTNP 

JEFOUJEBE EFM QBSBMFMPHSBNP  JNBHJOBSJP QVSP  ÎOàNP  JOUFSJPS EF VO DPOKVOUP  JOUFSWBMP  BCJFSUP  BDPUBEP  DFSSBEP  FODBKBEPT  5FPSFNB EFM  OP BDPUBEP  JOWFSTP BEJUJWP  JOWFSTP CJMBUFSBM  JOWFSTP MBUFSBM B EFSFDIB  JOWFSTP MBUFSBM B J[RVJFSEB  JOWFSTP NVMUJQMJDBUJWP  JTPNPSàTNP    

L MÎNJUF EF VOB GVODJÓO  MFZ EF DPNQPTJDJÓO JOUFSOB  MFZ EF TJHOPT   MFZ EF USJDPUPNÎB   MPHBSJUNP 

N OBUVSBM 

M NÃYJNP  NÃYJNP DPNÙO EJWJTPS  NÎOJNP  NÎOJNP DPNÙO NÙMUJQMP  NÙMUJQMP  NPEVMBUJWB 

±OEJDF BMGBCÊUJDP

NVMUJQMJDBDJÓO  NVMUJQMJDJEBE EF VO DFSP 

N OÙNFSP DPNQMFKP GPSNB USJHPOPNÊUSJDB  NÓEVMP  DPNQVFTUP  QSJNP  SFBM QPTJUJWP  OÙNFSP e  OÙNFSPT DPNQMFKPT  EF 'FSNBU  FOUFSPT   OFHBUJWPT  QPTJUJWPT  JSSBDJPOBMFT  OBUVSBMFT  SBDJPOBMFT  QPTJUJWPT  SFBMFT  OPUBDJÓO σ 

O PQFSBDJÓO CJOBSJB  PSEFO UPUBM 

P QBSUF FOUFSB  QBSUF GSBDDJPOBSJB  QBSUF JNBHJOBSJB  QBSUF SFBM  QFSNVUBDJPOFT  QMBOP DBSUFTJBOP  QPMJOPNJP  QPMJOPNJPT

QSPEVDUP EF  TVNB EF  QSJNPT SFMBUJWPT  QSJODJQJP EF JOEVDDJÓO  DPNQMFUB  NBUFNÃUJDB  QSJODJQJP EFM CVFO PSEFO  QSJODJQJP GVOEBNFOUBM EFM DPOUFP  QSPQJFEBE DBODFMBUJWB    EJTUSJCVUJWB  VOJGPSNF  QVOUP BJTMBEP  QVOUP EF BDVNVMBDJÓO  QVOUP àKP  QVOUP GSPOUFSB  QVOUP JOUFSJPS 

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