数学分析英文版-chapter4

CHAPTER4

LIMITSANDCONTINUITY

4.1INTRODUCTION

Thereaderisalreadyfamiliarwiththelimitconceptasintroducedinelementarycalculuswhere,infact,severalkindsoflimitsareusuallypresented.Forexample,thelimitofsequenceofrealnumbers{x},denotedsymbolicallybywriting

=A,

meansthatforeverynumber>0thereisanintegerNsuchthat

<whenevern≥N.

ThislimitprocessconveystheintuitiveideathatxcanbemadearbitrarilyclosetoAprovidedthatnissufficientlylarge.Thereisalsothelimitofafunction,indicatedbynotationsuchas

=A,

whichmeansthatforevery>0thereisanothernumber>0suchthat<whenever0<<.

Thisconveystheideathatf(x)canbemadearbitrarilyclosetoAbytakingxsufficientlyclosetop.

Applicationsofcalculustogeometricalandphysicalproblemsin3-spaceandtofunctionsofseveralvariablesmakeitnecessarytoextendtheseconceptstoR.Itisjustaseasytogoonestepfurtherandintroducelimitsinthemoregeneralsettingofmetricspaces.Thisachievesasimplificationinthetheorybystrippingitofunnecessaryrestrictionsandatthesametimecoversnearlyalltheimportantaspectsneededinanalysis.

Firstwediscusslimitsofsequencesofpointsinametricspace,thenwediscusslimitsoffunctionsandtheconceptofcontinuity.

4.2CONVERGENTSEQUENCESINAMETRICSPACE

Definition4.1.Asequence{x}ofpointsinametricspace(S,d)issaidtoconvergeifthereisapointpinSwiththefollowingproperty:

Forevery>0thereisanintegerNsuchthat

d(x,p)<whenevern≥N

Wealsosaythat{x}convergestopandwewritex→pasn→∞,orsimplyx→p.IfthereisnosuchpinS,thesequence{x}issaidtodiverge.

NOTE.Thedefinitionofconvergenceimpliesthat

x→pifandonlyifd(x,p)→0.

Theconvergenceofthesequence{d(x,p)}to0takesplaceintheEuclideanmetricspaceR.

Examples

InEuclideanspaceR,asequence{x}iscalledincreasingifx≤xforalln.Ifanincreasingsequenceisboundedabove(thatis,ifx≤MforsomeM>0andalln),then{x}convergestothesupremumofitsrange,sup{x,x,...}.Similarly,{x}iscalleddecreasingifx≤xforalln.Everydecreasingsequencewhichisboundedbelowconvergestotheinfimumofitsrange,Forexample,{1/n}convergesto0.

If{a}and{b}arerealsequencesconvergingto0,then{a+b}alsoconvergesto0.If0≤c≤aforallnandif{a}convergesto0,then{c}alsoconvergesto0.theseelementarypropertiesofsequencesinRcanbeusedtosimplifysomeoftheproofsconcerninglimitsinageneralmetricspace.

InthecomplexplaneC,letz=1+n+(2-1/n)i.Then{z}convergesto1+2ibecause

d(z,1+2i)==asn→∞,

sod(z,1+2i)→0.

Theorem4.2.Asequence{x}inametricspace(S,d)canconvergetoatmostonepointinS.

Proof.Assumethatx→pandx→q.wewillprovethatp=q.Bythetriangleinequalitywehave

0≤d(p,q)≤d(p,x)+d(x,q).

Sinced(p,x)→0andd(x,q)→0thisimpliesthatd(p,q)=0,sop=q.

Ifasequence{x}converges,theuniquepointtowhichitconvergesiscalledthelimitofthesequenceandisdenotedbylimxorbylimx.

Example.InEuclideanspaceRwehavelim1/n=0.ThesamesequenceinthemetricsubspaceT=(0,1]dosenotconvergebecausetheonlycandidateforthelimitis0and0T.Thisexampleshowsthattheconvergenceordivergenceofasequencedependsontheunderlyingspaceaswellasonthemetric.

Theorem4.3.Inametricspace(S,d),assumex→pandletT={x,x,...}betherangeof{x}.Then:

Tisbounded.

pisanadherentpointofT.

Proof.a)LetNbetheintegercorrespondingto=1inthedefinitionofconvergence.TheneveryxwithnNliesintheballB(p;1),soeverypointinTliesintheballB(p;1),where

r=1+max{d(p,x),…,d(p,x)}.

ThereforeTisbounded.

b)SinceeveryballB(p;)containsapointofT,pisanadherentpointofT.

NOTE.IfTisinfinite,everyballB(p;)containsinfinitelymanypointsofT,sopisanaccumulationpointofT.

Thenexttheoremprovidesaconversetopart(b).

Theorem4.4.Givenametricspace(S,d)andasubsetTS.IfapointpinSisanadherentpointofT,thenthereisasequence{x}ofpointinTwhichconvergestop.

Proof.Foreveryintegern1thereisapointxinTwithd(p,x)1/n.Henced(p,x)→0,sox→p.

Theorem4.5.Inametricspace(S,d)asequence{xn}convergestopif,andonlyif,everysubsequence{xk(n)}convergestop.

Proof.Assumexn→pandconsideranysubsequence{xk(n)}.Forevery>0thereisanNsuchthatnNimpliesd(x,p)<.Since{xk(n)}isasubsequence,thereisanintegerMsuchthatk(n)NfornM. HencenMimpliesd(x,p)<,whichprovesthatx→p.Theconversestatementholdstriviallysince{x}isitselfasubsequence.

4.3CAUCHYSEQUENCES

Ifasequence{xn}convergestoalimitp,itstermsmustultimatelybecomeclosetopandhenceclosetoeachother.Thispropertyisstatedmoreformallyinthenexttheorem.

Theorem4.6.Assumethat{xn}convergesinametricspace(S,d).Thenforevery>0thereisanintegerNsuchthat

d(x,x)<whenevernNandmN.

Proof.Letp=limx.Given>0,letNbesuchthatd(x,p)</2whenevernN.Thend(x,p)</2ifmN.IfbothnNandmNthetriangleinequalitygivesus

d(x,x)d(x,p)+d(p,x)+=.

4.7DefinitionofaCauchySequence.Asequence{xn}inametricspace(S,d)iscalledaCauchysequenceifitsatisfiesthefollowingcondition(calledtheCauchycondition):

Forevery>0thereisanintegerNsuchthat

d(x,x)<whenevernNandmN.

Theorem4.6statesthateveryconvergentsequenceisaCauchysequence.Theconverseisnottrueinageneralmetricspace.Forexample,thesequence{1/n}isaCauchySequenceintheEuclideansubspaceT=(0,1]ofR,butsequencedosenotconvergeinT.However,theconverseofTheorem4.6istrueineveryEuclideanspaceR.

Theorem4.8.InEuclideanspaceReveryCauchysequenceisconvergent.

Proof.Let{xn}beaCauchysequenceinRandletT={x,x,...}betherangeofthesequence.IfTisfinite,thenallexceptafinitenumberoftheterms{xn}areequalandhence{xn}convergestothiscommonvalue.

NowsupposeTisinfinite.WeusetheBolzano-WeierstrasstheoremtoshowthatThasanaccumulationpointp,andthenweshowthat{xn}convergestop.

FirstweneedtoknowthatTisbounded.ThisfollowsfromtheCauchycondition.Infact,when=1thereisanNsuchthatnNimplies<1.ThismeansthatallpointsxwithnNlieinsideaballofradius1aboutxascenter,soTliesinsideaballofradius1+Mabout0,whereMisthelargestofthenumbers,…,.Therefore,sinceTisaboundedinfinitesetithasanaccumulationpointpinR(bytheBolzano-Weierstrasstheorem).Weshownextthat{x}convergestop.

Given>0thereisanNsuchthat</2whenevernNandmN.TheballB(p;/2)containsapointxwithmN.HenceifnNwehave

+<+=,

solimx=p.Thiscompletestheproof.

Examples

Theorem4.8isoftenusedforprovingtheconvergenceofasequencewhenthelimitisnotknowninadvance.Forexample,considerthesequenceinRdefinedby

x=1-.

Ifm>nN,wefind(bytakingsuccessivetermsinpairs)that

=<,

so<assoonasN>1/.Therefore{x}isaCauchysequenceandhenceitconvergestosomelimit.Itcanbeshown(seeExercise8.18)thatthislimitislog2,afactwhichisnotimmediatelyobvious.

2.Givenarealsequence{a}suchthatforalln≥1.

Wecanprovethat{a}convergeswithoutknowingitslimit.Letb=.Then0bb/2so,byinduction,bb/2.Henceb→0.Also,ifm>nwehavea-a=;

henceb(1+)<2b.

Thisimpliesthat{a}isaCauchysequence,so{a}converges.

4.4COMPLETEMETRICSPACES

Definition4.9.Ametricspace(S,d)iscalledcompleteifeveryCauchysequenceinSconvergesinS.AsubsetTofSiscalledcompleteifthemetricsubspace(T,d)iscomplete.

Example1.EveryEuclideanspaceRiscomplete(Theorem4.8).Inparticular,Riscomplete,butthesubspaceT=(0,1]isnotcomplete.

Example2.ThespaceRwiththemetricd(x,y)=maxiscomplete.

Thenexttheoremrelatescompletenesswithcompactness.

Theorem4.10.Inanymetricspace(S,d)everycompactsubsetTiscomplete.

Proof.Let{xn}beaCauchysequenceinTandletA={x,x,...}denotetherangeof{xn}.IfAisfinite,then{xn}convergestooneoftheelementsofA,hence{xn}convergesinT.

IfAisinfinite,Theorem3.38tellsusthatAhasanaccumulationpointpinTsinceTiscompact.Weshownextthatxn→p.Given>0,chooseNsothatnNandmNimpliesd(x,x)</2.TheballB(p;/2)containsapointxwithmN.ThereforeifnNthetriangleinequalitygivesus

<+=,

soxn→p.ThereforeeveryCauchysequenceinThasalimitinT,soTiscomplete.

4.5LIMITOFAFUNCTION

Inthissectionweconsidertwometricspaces〔S,〕and(T,),whereanddenotetherespectivemetrics.LetAbeasubsetofSandletbeafunctionfromAtoT.

Definition4.11IfpisanaccumulationpointofAandif,thenotation

,(1)

isdefinedtomeanthefollowing:

Forevery>0thereisa>0suchthat

whenever.

Thesymbolin(1)isread“thelimitoff(x),asxtendstop,isb,”or“f(x)approachesbasxapproachesp.”Wesometimesindicatethisbywritingas.

Thedefinitionconveystheintuitiveideathatf(x)canbemadearbitrarilyclosetobbytakingxsufficientlyclosetop.(SeeFig.4.1.)WerequirethatpbeanaccumulationpointofAtomakecertainthattherewillbepointsxinAsufficientlyclosetop,with.However,pneednotbeinthedomainoff,andbneednotbeintherangeoff.

NOTE.Thedefinitioncanalsobeformulatedintermsofballs.Thus,(1)holdsif,andonlyif,foreveryball,thereisaballsuchthat∩Aisnotemptyandsuchthat

wheneverx∩A,.

Whenformulatedthisway,thedefinitionismeaningfulwhenporb(orboth)areintheextendedrealnumbersystemRorintheextendedcomplexnumbersystemC.However,inwhatfollows,itistobeunderstoodthatpandbarefiniteunlessitisexplicitlystatedthattheycanbeinfinite.

Thenexttheoremrelateslimitsoffunctionstolimitsofconvergentsequences.

Theorem4.12.AssumepisanaccumulationpointofAandassumebT.Then

,(2)

if,andonlyif,

(3)

foreverysequenceofpointsinA–{p}whichconvergestop.

Proof.If(2)holds,thenforevery>0thereisasuchthat

whenever0<.(4)

NowtakeanysequenceinA–{p}whichconvergestop.Forthein(4),thereisanintegerNsuchthatnNimplies<.Therefore(4)impliesfornN.andhenceconvergestob.Therefore(2)implies(3).

Toprovetheconverseweassumethat(3)holdsandthat(2)isfalseandarriveatacontradiction.If(2)isfalse,thenforsome>0andevery>0thereisapointxinA(wherexmaydependon)suchthat

0<but.(5)

Taking=1/n,n=1,2,…,thismeansthereisacorrespondingsequenceofpointsinA-{p}suchthat

0<<1/nbut.

Clearly,thissequenceconvergestopbutthesequencedoesnotconvergetob,contradicting(3).

NOTETheorems4.12and4.2togethershowthatafunctioncannothavetwodifferentlimitsas.