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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.
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