Introduction to Real Analysis INTRODUCTTIOOR NE ALA NALYSIS ThiErddi tion RoberGt.B artle DonalRd. S herbert EasteMrni chigUanni versYiptsyi,l anti UniversoifIt lyl inUoribsa,n a-Champaign JohWni ley SonsI,n c. & ACQUISITIEODNI TOR BarbHaorlal and ASSOICATEED ITOR SharPorne ndergast PRODUCTIEODNI TOR KenS antor PHOTEOD ITOR NicoHloer lacher ILLUSTRATIOCNO ORDINATOR GenAei ello Thibso owka ss eitnT imeRso mabny E igenCtoymppeo siatnopdrr si,n atnebddo unbdyH amilton PrintCoimnpga nTyh.ce o vwearsp rinbtyeP dh oenCioxlC oorr poration. Thibso oiksp rinotnea dc id-pfarpe§eer . Thpea pienrt hbioso wka sm anufacbtyua mr ieldwl h osfeo rmeasnta gempernotg rianmcsl suudset ayiineeldd harvesotfii ttnsig m berlSaunsdtsay.ii neehldad r vesptriinngce inpsluetrshe at htne u mbeorfts r eese acyhe ar CUI doenso etx cetehdae m ouonfnt e wg rowth. Copyr©i g2h0t0J 0o hWni leyS onIsn,cA .lr li grhetsse rved. & Nop arotft hpiusb licmaatybi eor ne prodsutcoeriden,adr etriseyvsatolert m r ansmitted ina nfyo romrb ya nmye anesl,e ctrmoencihca,np ihcoatlo,c orpeycionrgds,ic nagn,n ing oro therweixsceea,psp t e rmiutntdeSeder c t1i0oo7nr 1 0o8ft h1e9 7U6n itSetda tes CopyrAicgthw,ti theoiuttth heperr iworri tpteernm isosfti hoPenu bliosrha eurt,h orization thropuagyhm eonftt h aep proppreira-tcfeoe pteyot hCeo pyrCilgehta rCaennctee r, 22R2o sewoDordi vDea,n veMrAs 0,1 92(35,07 85)0 -8f4ax0( 05,0 785)0 -4470. Requetsott hsPe u blifsohpree rr missiobnea sdhdoruetlsodst hePede rmisDseipoanrst ment, JohWni leyS onIsn,c 60.5, T hiArvde nuNee,w Y ork,1 0158-0012, & NY (12)28 50-16,1f0 a(x2 1825)0 -6E0-0M8a,i l: [email protected]. LibroaCfryo ngCraetaslso igPunib nlgi cDaattai:o n BartRloeb,eG ratr dn1e9r2,7 - Introdutcor teiaaonlna lyRsoibse Gr.Bt a rtDloen,a Rl.dS, h erb-e3rrted.d . / p. cm. Inclubdiebsl iogrraepfheirceaannlicd ne dse x. ISB0N -4137-214(8a-16pka .p er) 1.M athemaatniaclayls2 i.Fsu .n ctoifor nesav la riables. Sherbert,R .1D,9o n3a5.l-I dIT .i tle. 1. QA300.B2209040 515-dc21 993-8129 CIP A.M .S .C lassifi2c6a-t0i1o n Prinitnte hdUe n ited oSfAt maetreisc a 20 19 18 17 16 15 14 13 12 II PREFACE The orsfet aaunyldasi yliis sn d ifsopapre r nossigpbrelacsedtt uiuoavdpfteeu eo nr rte appmlaitehsdIe.a tm l ahstagoisr cv eaafltoau rneu y n dersgtruwadhdweouin astth ee s tgobo e yornodum tatinhnieepo uffrol mauttsloioa oslsntv saep no rdbalbreedmc sa,u se idte ovpteshal eb ittlohi ditenydk u acntaimlvayetzlheyes ,mii atotuanianesctdx,a t le nd idteaona escw o ntreexyctee.man artts ih,hce asbms ea ctvoamlieumn aa banelryae s , In incelcuodnaionnmmdgai ncassgc eiamweseenn altctsplhe h e y ssciiceeannlgc iense,e ring, ancdo mspcui.tOe eugnrroc i atesplo r oavanic dceer sesaispboalntceaee,bxdi lt nyb ook thfuen dacmoennactnteadepl ct hsorn feia aqnluaf elosyr s iistnsth uaedrsTeeehna itsss. boiodske sfiosgrtn uewddhhe oans vttesuc dailaecisdiut tr sl a udsi ptrieoisnnea nltleyd the SUtenWasiht.tsi etlduefi d nteidhbns ot coshk a l,l eeoxnupgreii trsnhis gaeestnr cieo u stuadttehl nieatsvrs ee lc afpouamflba llsyett hemera itpenrrgei hsaeelrn et.e d Thet wefiodr isotttfi h boionwsose kv r eewr eryle lc aenwidehv aetvda,epk aeinn s tmoa itnhsteaas mipenai nurdsi etr a-pfprIripnoer anecdtphlhae.yird siiw ntehgi a ovne, exaemvisenerecaydtn sidoeoe ftnxc eisrstreesa,sm olamireng eupdmr eoanvft iesdw,e d neewx ammopvleedst ,oc tpneoier lcwtos ac iaanntm diar odenevs Ei,xs cifeootprnht se . new Cwhhadipectwaheil rtts hgh ee neRriaelmiaznenhd a nivonaettd edgerda l, we 10, muncehmw a atWleh.rti ihlieemsr o emr aet tehrcaiabnalnce o vieonrn seeed m ester, instmrawuyict sutohs o er tscio eacphrssot najoiernocs er tx cpstrr raeoasd siit.g nments Iitds e stihtrahsatetb ulhdeahe vasnedot e m xep osubruwete tadoso s pnuromoteo fs, thtabote ctaThsoee .ps roomvefi odhree ilsanptn uadplerynozottoifshfn e esgmo sr, wien calanup dpoee"n nL doigxPi rcot ohdafainstsd" tc ouspsuiscciehsms p ialosin csa,t quaenrntsei,gf, aci toinsotinrstasa,inpd vdoie ffteyropepfenr stoW oef sh.at vhee kept disciunsfsoairvmoboaneil cd mo timoirit nenh dge teocffho nrlimocWagaelil c .d etails feetlhi atatm oiursse ee xfpueltrl oie hearnon twcco oe n psrtobrfioyu rfcwsstat t ching antdhd eonti hnbargyne aadbitoneugct ho pnfri oqoufe.s Weh avea ma eddolipeuotvgmfeee dln ecroanlstiiohtsurytgt ehhbnoeotuw olteky : prerseetsnhuatalrg ttee s ne enrotacuolog c vhae tsrhae acstt a iursabeulw,tlen y osd tto r ive for maxiImtnuhm meaw iepgn re,on focermetr ehapdela irttttoyihg .cee u n.lTe ahrrua sl wceo ncsoinodtufenisurcn tuoionop anencsndl ionsteieddnr e vbtauwaltaeis r l e, careful tpor epsreotnohctfaar stne abdaeid latypaom t oerde sgiet.(nu ICeanhrt aaiplot ner 11 we ptaarkatedi vcaoutnflhata epar pg .reW)o eb a eclthiheiaivstme p itotpor rt oavnitd e stuwdimetanhent xysa tmaopit ldhei estnm h u enidre r,as nwtdeha ancvdoeim npgiled raetxhtelerin osesftxi sevt recoc h iastlehlsWee hmnwi.gel el e ed raoov upetr ioanosef s exiesrweceds no,o t trt ayot btraeibvnryi e tlyed giaffiptcriuotnloogtf stc hies eesx.e r Howeivsneo rom,tfe hl eas teescrwt, edi oo ndborawem nao kde drfiaifetcxeuiellsrytec inast eoq ousfet necpes . ICnh apwtepe rreab s reinetf o tfshn ueom taminanodron ytsf a ostreia tonsnd s fnucttihwo1aue nts As d e i.s coMufas tshiIeomnnad ituasiclg ctsiaiosvlo ie ninn nc,de u ctive praorofeifrqssu eWe ean ltislnoycas .l h usoderecot ftn ii cnooinut anenti,dan bfslieent ist.e Wer ecotmhmaectnh datb pchetoi evsqre u rieocudrks laebysda, c kmgaratoleu,rn id retluarnatniseen rcg e ssary. v PREFACE vi Chap2tp eersre tnhpters o tpieoerfts h ree naulm bseyrs teThmfie r tswtso e ctions lR. dewailtt hhAe l gebarnOadri dcPe rro ipeeasrnpt dr ovsiodmpeer t aicicwner iptrionogf s ofe lemernetsaruTylh cters u.Cc oimapll sesPt reonpeertyi nS iesc t2ig.aoi3stvn he en SuprePmruomp aenridttr ymsa,i ficaardteii osnctsuh srsoeudtg hhreeo muati onftd heirs chapter. InC hap3tw eegr i avt eh ortoruegahot fms eenqctue eisnnIR a ntdh aes socliiamtietd concTehpmeta st.e roifta hlge r iesai tmepsotr tance; fofriitnrtuda n tahteerl y, students natuarlatlhi ottua gksheo smt ei fmoetr h etmob ecome fulltyot h uaesc oecf u stomed €. Int hnee Swe ct3i.wo7eng, i avb er iienft rotdioun cftsiienorinsit otee hs ta,hti i msp ortant topwiinclo lbt eo mitdtuteeaod s horotfta igmee . Chap4to enlr i moifftn usc tiaonCndhs a p5to encr o ntifnuutnoicuocsno sn stitute thhee aorftt hb eo oOku.dr i scuosfls iimaoinntcd so ntirneulhiieteaysvo intl huyes oef seqcueeasnn,td h cel oselya ppparoroftaa hclehlcs ehela p rteeoirrnescfts h uen derstanding oft heesses etnotpiTiahcflesnu .d amepnrtoaipleeo srfc t ontifnnuucotu(isooi nnn st ervals) ardei sciunSs escet5di .aon3n5d4 .T.h neo toifao" ng auigisen "t roidnSu eccet5di. o5n anuds tedgo i avlet ernpartoioovftfeh s e set ipMeroson.po etfronuncet iarodeni ss cussed inS ect5i.o6n. Thbea stihceo otfrh yde e riivgasit viievtnneh f ei prasortCft h ap6tT.eh rii msp ortant matersitaaln edixascr tedhp,wat et h avues eard e soufCl atr athteogo idsvoierm yp ler proootffh sCe h aRiunla ent dh Ien vieoTrnhs eeomTr.h ree maionftd hceihrsa pctoenrs ists ofa plpicaotfti hoen sVM aleuaTenh eoarnemdma bye e xplaostr iepmdee r tmsi. Chap7td,ee ra lwiinttghhR ei emianntne hgarbsae lec,no mplreetveiilsntye h di s editRiaotnth.he arin n troudpupcaeinrnd g lionwtee(rga rwsae dl isid nt hper ieovus editwieho endrsee)f ,ti hinene t eagaslr iaomlRfi ite msaunmnTs h.hi asts ha ed vantthaagte iitcs o nesnwitis ttthhse t ud'fie rnestxtsp otstouh irene t eigcnra allca unildanu p slp ications; siniictnse o d te penodnoe rndpter rot pieieprtse ,r miimtmsed gieantezera attlicoioo nm plex anvde ctorf-nuvcatlituohensadstt u dmeanyet nsc ouinnlt aetcreo ru rCsoetnsr.at roy populart holipisima nipitpo rnion,sao c hm doirfefit chualnot r tdaheperp c rhoI.aat l isso consiwsittehgn ettnl heierzR aeide mianntnet ghiradsatil s ciunds esteiadnCi hla p1t0e.r Sect7i.go4in va be rsi deifs cuosfts hfieaoi mnl niuamre rmiectaholofc da sl cutlhaet ing inteogfcr oanlt ifnnuucotuiso ns. Sequeonffcn uecst iaonundns oi rfcmo nveragrdeein scceiu nts hsfieer dts wtso e c tionCsh aop8ft,a e ntrdh bea stirca nscfenuncdteianrotpenauo sltn a f ifromu ndiant ion Sect8i.ao3nn8 d . b4yu siunngi fcoorngmv eenrCchea.p9 ct oemrp loeudtrie ssc uosfs ion infinsietrCeih easp.8t a enr9dsa re intirmipnos,rai tncatdanh ltaellyyss hoo w thhoew mateirtnih eaeal r clhiaeprct aebnrea s pp lied. Chap1ti0ecs ro mplneetwiei;tlas py r eseonftt hagete inoenrR aileimziaenndtn e gral (somectailtmlhe"eesHd e nstocko-rtKh "uegr azuwgeei"lI "wti i nbltenle e gtwrmo aa ln)y. readaenrwdse t, h ink bteha emya ztwehidsal utlca hna pparmeinnmtoolryd ifiocfa tion thdee finitthiReoi ne moiafnn tnec garna tlloa en ia ndt etghiramsato l r gee netrhatalhn e LebeisngtueeWg ebr eallti.he tavhteri esl anteiawvp epalrcytoh io n tegtrhaetiobisroo ytn h accesasneidxb clietta oin nygwo hnhoea sst utdhibeea dsR iice mianntn.e g ral Thfei nal Cdheaawplitstte ohrp olcoogniccEeaaplrt plsri.oe gorif vse inn tfeorrv als 11 areex tetnoadm eodra eb stsreatcFttoie rnx ga.m, tp hlceeo ncoefcp otm tpanceisgsis v en proepmeprah sainsmd e tsrpiacc eisn tuarcroeTeddh .ic sh apwtieblrelv eruys effourl studceonnttsit nogu riandcguo autriesnm e ast hteimcas. Thrgohuotuhbteo owkeh avpea miodr aet tetnott oipofinrc onsmu merainlcayasli s anadp pirmoaxttihoetnoh rayun s uiWasel h .a ve sdobo enceao ufts hieem porotfa nce theaesraeas n,td so hotwhr aetaa nlas liiysns o mte raenle yxr eciisanbe s ttrhaocutg ht. PREFACEv ii We hparvoev riadtelhdee nrgl tihosyfet xsce irsseosme,ea asnysd o mceh allenging. Weh ave p"rhnoitvfsio"dr e dom fta hneyes xceei rstehose ,sl tpu dgeesnttt astr otweaad r d soluotrti cooh net chke" iarn s.wM eorr"e cosmoplluoetfati leom neosvs eter xcyei rasree givienns eap aIrnastte'rs Mu acntuowarhli,ica shv aitltoae balceuh peorrnes qs utteo the publisher. Iit ss aat isefxypienrtgios e enheco ewt hmea thematicoaftl h setm uadteunrtisty incsreaesna hdo wt hset udgernatdsul aelarntl owy o rcko mforwtiatcbhol nyct ehpatts initsieaelmsleomyd y sitoeuBrsu.t itnsho de oruetb htaal too thf a rd iwrsoe rqkuo inr ed thpea orfbt o tthhse t udaentndht tese ac.h ers Ino rdteeorn rtihchehi storicaolf bpthoeeor ks,p ewcbetr iibiveinefoc glruadpeh ical sketocfsh oemfsmea o umsa themawthicoco inatnrtsito bh uaitraseeW. de a rpea rticularly indetboDt re.d PMautlrdiocfwkonp rer yo viudswi inthghip sh otoogfPr raeopssfho rs HenstoKcuzkrw aeWniedla .lt shoa JnokhW ni leSyo nfso r obpthaoitnoigonrfga phs & thoet hmeart heimaantsi.c Weh avree cemiavnehyde lpcfoumlm efonrmtc so lleaatagw uiedvseae rtioyfi n stitwuhthoia ovtneas u fgorhmte areldiietrai noldni skt ehbdeo oekn outgoeh x epsrs thoepiri naiboohnuosttw io m prioWtve.ea ppretchireaeitmrea a rnksdsu ggeesvteino ns, thowuegd hin doa tl wfaoylstl hoeawid rv Wiect eh.at nhkef moc ro mmuniwciautthsi ng anwdi sthh ewme ilntlh eeinrd eatvioom rptsah rceth alalneednx gcei toeflme earnnitn g reaanlas liaysn" dr emaalt"h teimIcaitsos .u h ro pteht ahtwe iyfil nltd h niesew d ietvieonn more htehltaphnefe au rlol nieesr. Fberuary 241,9 99 RobeGr.Bt a rtle YspialnUaadrnnb tai DonaRl.Sd h erbert THE GREEKA LPHABET A ex Alpha N Nu v B fJ Beta � � � Xi Gamma Omicron r y Delta 0 0 Pi � /) Il 7r E Epsilon P Rho Z e Zeta p Sigma � I; a H Eta T Tau 1'/ r Theta Upsilon e () 1 v I Iota Phi <I> ({J K Kappa Chi K X X Lambda 1/1 Psi A ).. \II M Mu Omega J.i- Q w To our wives, Carolyn and Janice, with our appreciation for their patience, support, and love. CONTENTS CHAPTER1 PRELIMINRAIES 1 1.S1e atnsFd u nct1i ons· 1.M2a themIantdiucca1tl2i on 13. FinitIen fianSniedtt1 es6 CHAPTER2 THER EALN UMBERS 22 2.T1h Ael gebarnOadri dcPe rro peorfIRt 2i2e s 2.A2b soVlaluutee anLdi n3Re1e al 2.T3h Ceo mpelneePtsrso poefIRr t3y4 2.A4p plicoaftt hSieuo pnrse Pmruomp e3r8 ty 2.I5n ter4v4 als CHAPTER3 SEQUENCEASN D SERIES5 2 3.S1e queannTcdhe esLi irm i5t3s 3.L2i mTihte or6e0m s 3.M3o notSoenqeu e6n8c es 3.S4u bseqaunetdnh Bceoe lnszo a-eWierTshteroars7e5s m 35. ThCea ucChryi te8r0 ion 3.P6r opDeirvleyrS geeqnute 8n6c es 3.I7n dturcottioIon nfi nSietrei8 9e s CHAPTER4 LIMITS 96 41. LimoifFtu sn ct9i7o ns 4.L2i mTihte or1e0m5s 4.S3o mEex tseinoontfs hL ei mCiotn ce11p1t C.HAPTE5R CONTINUOUFSU NCTIONS 119 5.C1o ntiFnuunocuts1i 2o0n s 5.C2o mbinoafCto inotnisFn uunocuts1i 2o5n s 5.C3o ntiFnuunocutosinI o nntse r1v29a ls 5.U4n ifCoornmt in13u6i ty 5.C5o ntiannuGdia tuyg1 e4s5 5.M6o notoInvnee rFasunend c t1i4o9n s ix