Introduction to Real Analysis - Columbia University

1 Intro ductiontoRealAnalysisJoshuaWilde,revised byIsab elTecu,TakeshiSuzukiandMar aJos Bo ccardiAugust13,20131 SetsSetsarethebasicob ,theyaresobasicthatthereisnosimpleandpre cisede osesitsu cestothinkofasetasacollectionofob :A={a,b,c}meansthesetAconsistsoftheeleme ntsa,b, erofelementsinasetcanb e niteorin ,thesetofallevenintegers{2,4,6,..}isanin :a A a {}or .Notethatanystatementab erationsTheunionoftwosetsAandBisthesetco nsistingoftheelementsthatareinAorinB(ori nb oth).ItisdenotedA ,ifA={1,2,3}andB={2,3,4}thenA B={1,2,3,4}. ,ifA={1,2,3}andB={2,3,4}thenA B={2,3}.Supp ,denotedACisthesetofallelementsinXthatar enotcontainedinA:AC={x Xsuchthatx / A}.TheCartesianProductoftwosetsAandB,den otedA B,isthesetofallp ossibleorderedpairswhose rstcomp onentisanelementofAandwhosesecondcomp onentisanelementofB:A B={(a,b)suchthata Aandb B}. ,Countable,andUncountableSetsAsetAissett ob e niteifthereexistsabijective("one-to-onea ndonto")functionfmappingfromaset1,2.

2 , nitenumb erofelements-toeachelementwecanassignexa ctlyonenumb erfrom1,.., niteifitisnot nitelymanyelementsbutthattheoreticallywe couldcountthemall(ifwehadin nitetimetodoso).AsetAisuncountableifitis neither ersQandthesetofintegersZareb ,butwecannotdomuchwiththemineconomicsunl esswede neanadditionalstructureonthem-thenotiono fdistanceb etweenelementsinaset, nitionAsetXissaidtob eametricspaceifwithanytwop ointspandqofXthereisasso ciatedarealnumb erd(p,q) (p,q)>0ifp6=q,andd(p,q) = 0ifp=q; (p,q) =d(q,p); (p,q) d(p,r) +d(r,q)foranyr XAnyfunctionwiththesethreeprop ortantmetricspacesweencounterineconomics aretheEuclideanspacesRn,inpartic-ularthe realnumb ,themostcommonlyuseddistanceistheEuclide andistance,whichisde nedasd(x,y) =||x y||= N i=1(xi yi) ,ifyouareonthesouth-westcornerofacityblo ckandyouwanttogotothenorth-eastcorneroft hesameblo ck,youmusttraveleastoneblo ckandnorthoneblo cks,whereastheEuclideandistanceis 2blo ede nedmathematicallyasd(x,y) =N i=1|xi yi|.

3 NitionsEquipp edwithadistancedwecande nethefollowingsubsetsofametricspaceX(you cansimplythinkofdastheEuclideandistancea ndofXasRN):Op enandClosedBallsThesetB(x,r) ={y X:d(x,y)< r}iscalledtheop enballB(x,r) (x,r) ={y X:d(x,y) r}iscalledtheclosedballB(x,r) enball,aclosedballcontainsthep ointsoftheb oundarywhered(x,y) = eled enballissimplyanop eninterval(x r,x+r), {y R:x r < y < x+r},andaclosedballissimplyaclosedinterval(x r,x+r), {y R:x r y x+r}.Op enandClosedSetsAsetU Xisopenif x Uthereexistsr >0suchthatB(x,r) :Asetisop enifforanyp ointxinthesetwecan ensetisasetthatdo esnotcontainitsb oundarysinceanyballMathCamp3aroundap ointontheb oundarywillb ,theinterval(0,1)isop eninRsinceforanyp ointxin(0,1),wecan ndasmallintervalaroundxthatisalsocontain edin(0,1).AsetU Xisclosedifitscomplimentisop nitionisthatasetisclosedi sequences{xk}withxk U kand{xk} x,thenx oundary(sincethecomplementofthatsetdo esnotcontaintheb oundaryandisthusop en).

4 Thede nitionusingsequencessaysthatifasequence{ xk}getsarbitrarilyclosetoap ointxwhilestayingintheclosedsetthenthep ointxalsohastob ,theinterval[0,1]isclosedinRsinceitscomp lement,theset( ,0) (1, ),isop eop en( (0,1)),closed( [0,1]),neither( (0,1])orb oth({},R)!ProblemShowAsetU Xisclosedifandonlyif sequences{xk}withxk U kand{xk} x,thenx Xisboundedif r >0andx XsuchthatU B(x,r).Thisisaneasyone:Asetisb oundedifwecan titintoalargeenoughballaroundsomep oundedifnomatterhowlargewecho osetheradiusoftheball,thesetwillnotb RNiscompactifitisclosedandb tsintoaballandcontainsitsb ,thede nitionofcompactsetisdi erent, nitenumb erofp oints{x1,..xn}, xi RN,ap ointz RNisaconvexcombinationofthep oints{x1,..xn}if RN+satisfying Ni=1 i= 1suchthatz= Ni=1 ,theconvexcombinationsoftwop ointsinR2formthelinesegmentconnectingthe twop ,x2 X,then x1+ (1 )x2 X,where [0,1].Thesecondde nitionsaysthatasetisconvexifyoucandrawas traightlineb etweenanytwop ,ifyoupickanytwop ointsintheunitdisk, :(1,0)and( 1,0)areb othontheunitcircle,butthelineconnectingt hemgo esthrough(0,0), , (Jensen'sInequality)Letf:RN neftob eaconvexfunctioniff( x+ (1 )y) f(x) + (1 )f(y),0 1foreveryx,y (N i=1 ixi) N i=1 if(xi)whenever i 0foralliand Ni=1 i= enballsareop ensetsisop niteintersectionofop ensetsisop nitionAsequenceisanassignmentoftheelemen tsinsomesettothenaturalnumb eledfromzero(orone)toa nitenumb erorin nity:Finitesequence:{xn}Nn=0={x1,x2.)}

5 ,xN}In nitesequence:{xn} n=0={x1,x2,..}Examples:{xn} n=0={1,1,2,3,5,8,..}{xn} n=0={1,0,1,0,..}{xn} n=0={1,12,13,14,..}Sequencescanalsob ede :a(n) =n2 {xn} n=0={0,1,4,16,..}b(n) =nn+ 1 {xn} n=0={0,12,23,34,..}c(n) = ln(n) {xn} n=1={ ln(1), ln(2),..}d(n) =enn {xn} n=1={e,e22,e33,..}SubsequencesGivenasequ ence{xn},considerthesequenceofp ositiveintegers{nk}suchthatn1< n2< n3< ..Thenthesequence{xnk}iscalledasubsequen ceof{xn}.Example:Let{xn}={1n}and{nk}b ethesequenceofprimenumb {xnk}={1,13,15,..}. ertiesofSequencesinRWe rstlo okatsequencesinonedimension, ertiesandde nitionsgeneralizeeasilytosequencesofhigh erdimensions, ,sinceasequenceinRNcanb {xn} n=0inRisboundedaboveifthereexistsanumb erMasuchthatallelementsofthesequencearel essthanMa:xn Ma n {xn} n=0inRisboundedbelowifthereexistsanumb erMbsuchthatallelementsofthesequenceareg reaterthanMb:xn Mb n oundedb othab oveandb erMawhichisanupp erb oundofasequenceiscalledtheleastupperboun dorsupremum,whilethelargestnumb erMbwhichisalowerb oundofthesequenceiscalledthegreatestlowe rboundorin :Thesequence{xn} n=1={0,1,0,2,0,3,0,4.

6 }haszeroasitsgreatestlowerb oundandhasnoupp erb ,thesequenceisb oundedb elow,butisnotb oundedsinceitisnotb oundedab xn xn+ xn< xn+ xn xn+ xn> xn+ ,Convergence,andDivergenceDe nitionAsequence{xn} n=1inRhasalimitL Ri foreach >0, K Z++suchthatifn K,then|xn L|< .Wewrite{xn} n=1 Lorlimn xn= ,theremustb eanumb erKsuchthatallelementsaftertheKthelementmustb eintheepsilonballB (L). , :Provethesequence{xn} i=1={1,12,13,..} of:Itsu >0, K Z++suchthatifn K,then|xn|< .|xn|< |1n|< 1n< 1< n n >1 .Cho ose > >1 >1 actuallyworks,justlet di erentnumb ,let = ,wecaneasilyseethatanynumb er1nwheren >1isgoingtob elessthan = =12,thenourKmustb >2,1n< = {xn} n=1 x,and{yn} n=1 y,then{xn+yn} n=1 x+ {xn} n=1 x,and{yn} n=1 y,then{xnyn} n=1 {xn} n=1 x,xn6= 0foranyn,andx6= 0,then{1xn} n=1 :Let{xn}b eab {xn}converges.(Pro ofnotonthesolutionsheetbutonWikip edia.) :Everyb oundedsequenceinRhasaconvergentsubsequen ce.

7 (Hint:Showthateveryb ) {xn} n=0inRNcanb eessentiallyviewedasavectorofNsequencesi nR:{xn} n=0= {(x1)n} n= {(xN)n} n=0 Let'sseehowsomeoftheprop ertiesab oveextendtomultidimensionalsequences:Ase quence{xn} n=0inRNisb oundedifthereexistsanumb erMsuchthatallelementsofthesequencearele ssthandistanceMfromtheorigin, (xn,0) M n onentsequences{(x1)n} n=0,..,{(xN)n} n=0isb {xn} n=1inRNhasalimitL RNi foreach >0, K Z++suchthatifn K,thend(xn,L)< .Inotherwords,theremustb eanumb erKsuchthatallelementsaftertheKthelementmustb eintheepsilonballB (L).TheoremAsequenceofvectorsinRNconvergesifandonlyifallthecomp {xn}={(1n,0)} {xn} (0,0).4 ContinuityHerewerevisitthede , ,letXandYb etwometricspacesendowedwithmetricsdXanddYresp ectively,andletfb (x) qasx porlimx pf(x) =qifforevery >0wecan nd >0suchthatdY(f(x),q)< forallxforwhichdX(x,p)< .qiscalledthelimitoffatthep ointp Xiflimx pf(x) =f(p).

8 Thisisequivalenttosayingthatforevery >0wecan nd >0suchthatdY(f(x),f(p))< forallxforwhichdX(x,p)< .fiscalledcontinuousifitiscontinuousatev eryp eacompactsubsetofRandletfb (X)iscompact.(You'llneedmorethanwecoverh eretoprovethis,sodon'ttryunlessyouknowwh atyou'redoing.) eacompactsubsetofRandletfb etheleastupp erb oundoff(X)andletMbb ethegreatestlowerb oundoff(X).Thentherearep ointsp,qinXsuchthatf(p) =Maandf(q) = ' of:f(X)iscompactbytheorem1,thereforeclos edandb nitionMaliesontheb oundaryoff(X).Sinceclosedsetscontainsthe irb oundaryp oints,Mahastoactuallyb einf(X).Thatmeansthereexistssomep ointpinXsuchthatf(p) = en,closed,neither,orb oth:1.{(x,y) : 1< x <1,y= 0}2.{(x,y) :x,yareintegers}3.{(x,y) :x+y= 1}4.{(x,y) :x+y <1}5.{(x,y) :x= 0ory= 0} enballsareop ensetsisop niteintersectionofop ensetsisop oundandtheleastupp erb ,provewhethertheyareconvergentordivergen t:1.

9 {xn} i=1={12,23,34,45,..}2.{xn} i=1={ 1,1, 1,1,..}3.{xn} i=1={ 12,23, 34,45, 56,..} {xn} n=1 xand{yn} n=1 y,then{xn+yn} n=1=x+ allthecomp {xn} n=1={(1,12),(1,13),(1,14),..}convergesto (1,0). {xn} n=1={(12,12),(23,13),(34,14),..}converge sto(1,0).