微积分讲义Chap 1 Completeness axiom of R

Chapter1Therealnumbersystem1.3.CompletenessaxiomofR1.16DefinitionLetREandE.(i).ThesetEissaidtobeboundedaboveifthereisanRMs.tMaforallEa.(ii).AnumberMiscalledanupperboundofthesetEifMaforallEa.(iii).AnumberSiscalledasupremumofthesetEifSsatisfiesthefollowingconditions(1)ifEaSa,,(2)ifMisanupperboundofEthenMS.RemarkThesupremumisalsocalledtheleastupperbound.1.17:ExampleIfE=[0,1],provethat1isasupremumofE.Proof.1.]1,0[,1Eaa.2.letMbeanupperboundthenMaforall]1,0[a])1,0[1(1M.Wederivetheresult.1.18:RemarkIfasethasoneupperbound,ithasinfinitelymanyupperboundsProof:.LetEbeasubsetofR.LetMaforallEa.ThenMisanupperbound.LetRbb,0thenM+bisalsoanupperbound.So,Ehasinfinitelymanyupperbounds.1.19.Theorem.LetEbeanonemptysubsetofR.ThentheleastupperboundofEisuniqueifitexists.Proof.Supposethat21&ssaretheleastupperboundsofE.Then21&ssareupperboundsofE.1221&ssss21ss.NotationThesupremumisalsocalledleastupperbound.WeusesupEtodenotethesupremumofnonemptysetE.1.20.Theorem[Approximationproperty]EEREsupand,,exists.ThenEaanisthere,0s.tEaEsupsup.Proof:.Supposetheconclusionisfalse.Thereisan0suchthat.,supEaEa.Esupisanupperbound.EEsupsup0Eas.tEaEsupsup1.21.TheoremIfNEhasasupremum,thenEEsupProof.LetsupE=s.ByApproximationproperty,thereEx0s.tsxs01.Ifsx0thenEEsupisobvious.Ifsxs01,thenEx1s.t001100xsxxsxx.1.1,0101xxNxx.2.1)1(1,0101ssxxsxxs.Itisacontradiction.EEsup●[CompleteaxiomofR]EverynonemptysubsetEofRthatisboundedabove,thenEhastheleastupperbound..1.22:[ArchimedeanPrinciple]NnbaRba0,,,s.tbna.Proof:1.Ifba,thentaken=1.2.Ifab,let};{bkaNkE.EE,1.Ekabk,Eisboundedabove.ByCompletenessofR,supEexists.baEEEEE)1(sup1sup)21.1Theoremby(suptaken=supE+11.23:Example.Let,.......}41,21,1{Aand,...}87,43,21{BprovethatsupA=supB=1Proof.1.1{;0}2nAnNorn11,,0,1,2,..2nxxn.1isanupperbound.LetMbeanotherbound..1sup1210AM2.};211{NnBnNnn,21111isanupperboundofB.LetMbeanupperboundofBToshow1M.Supposenot011MMByArchimedeanprinciple,thereexistsNnsuchthatMn11,Mn121forsomeNn.MMnnnnn212)1(1212211Misanupperbound.1M1supB●[Well-OrderPrinciple]ENE,Ehasaleastelement(ie.Eas.tExxa,)1.24.Theorem(Densityofrational)LetRba,satisfyab,thenthereisarationalnumbercs.tacb.Proof:LetNnabn,1(byArchimedeanPrinciple).1.Ifb0,let}.;{nkbNkEByArchimedeanPrincipleE.ByWell-OrderPrincipleEhasaleastelement,says0k..)..(1:0bnmeiEmkmLetnmq.Wemustshowthataqb.qbisobvious,nowweshowthataq....11)(00bqaaqqnknnkabba2.Ifb0,then0k,kisanaturalnumbers.tb+k0.Qcs.ta+kcb+kQkcQcbkcaie.Thereisarationalnumberbetweena&b.1.27.Definition.ERE,.1.siscalledalowerboundofEifExsx,.Inthecase,Eiscalledboundedbelow2.tiscalledthegreatestlowerboundofEif1.Extx,,2.IfMisalowerboundofEthentM.3.EisboundedifExMx,forsomeM0.(i.e.Eisboundedaboveandbelow.)●LetEbeasetofR.Wedefine};{ExxE.1.28.TheoremERE,.1.supEexistsinf(-E)existsinfactsupE=-inf(-E)2.infEexistssup(-E)existsinfactinfE=-sup(-E)Proof:1.supEexists.Nowweshowthat–supE=inf(-E).Showthat1.-supEisalowerboundof–E.2.ifsisalowerboundofEsEsup.1.EsupisanupperboundofEExExExEx,sup,supEsupisalowerboundof–E2.Supposethatsisalowerboundof-ESupposenotEsEssupsupontheotherhandsxExsx,Hence,-sisaupperboundofEBy1.&2,EEEsup)inf(&)inf(.Theproofofconverseissimilar.Remark.Thelargestlowerboundisalsocalledinfimum.Remark.ThecompletenessaxiomofRisequivalentto“Everynonempty,boundedbelowsubsetofRhastheinfimum”.1.29.Theorem.infinf,supsup,,ABABBARBAifBBinfandsupexist.Hence,BAABsupsupinfinfProof:1.supposesupBexists.,sup.,supAxBxBABxBxAisboundedabove&supBisanupperboundofABycompleteaxiomofR,.supsup&supBAA2.SpposethatinfBexists..,inf,infAxBxBABxBxAisboundedbelow&infBisanlowerboundofA.BycompleteaxiomofR,BAAinfinf&:inf.Def:sup,inf1.4Functions,countabilityandthealgebraofsets.DefinitionLetA&BbetwosetsofR.AfunctionfisarelationbetweenA&Bs.tfassignseachelementxofAtoauniqueByDefinitionBAf:fiscalled1-1if)()(yfxfyxDef:BAf:fiscalledontoifAxBy,s.tf(x)=yDefinition1.34:LetEbesasetofR.1.EissaidbefiniteifEorEnfNnn}...2,1{:&},.....3,2,1{s.tfis1-1&onto.2.EiscalledcountablyinfiniteifENf:s.tfis1-1&onto.3.EiscalledcountableifEisfiniteorcountablyinfinite.4.EiscalleduncountableifEisnotcountable1.35.Theorem.Theopeninterval(0,1)isuncountablePf:Suppose(0,1)iscountable.Thenthereisalistfor(0,1)says.....................................................................0............................................................................0.........0.........0321333231323222121312111nnnnaaaaaaaaaaaaaaaaLet.......0321xwhere{k10kk.1f,1ifkkkkaiakkkaxisnotinthislist(0,1)isuncountable1.37.TheoremBBA,iscountableAiscountable1.38TheoremnAAA,......,21arecountable.},:{:1NjAxxAAEjjjNjj.IfjAiscountableforENjiscountable.