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TeoLeePeng

MathematicalAnalysis

VolumeI TeoLeePeng

January1,2024

3.5.3TheTrigonometricFunctions

Chapter4IntegratingFunctionsofaSingleVariable

4.1RiemannIntegralsofBoundedFunctions

4.2PropertiesofRiemannIntegrals

4.3FunctionsthatareRiemannIntegrable

4.4TheFundamentalTheoremofCalculus

4.5IntegrationbySubstitutionandIntegrationbyParts

5.1LimitSuperiorandLimitInferior

5.2ConvergenceofSeries

5.3RearrangementofSeries

5.4InfiniteProducts

5.5DoubleSequencesandDoubleSeries

6.2UniformConvergenceofSequencesandSeriesofFunctions

6.5TaylorSeriesandTaylorPolynomials

6.6ExamplesandApplications

6.6.1TheIrrationalityof

6.6.2TheIrrationalityof

6.6.3InfinitelyDifferentiableFunctionsthatareNon-Analytic548

6.6.4AContinuousFunctionthatisNowhereDifferentiable

6.6.5TheWeierstrassApproximationTheorem

Preface

Mathematicalanalysisisastandardcoursewhichintroducesstudentstorigorous reasoningsinmathematics,aswellasthetheoriesneededforadvancedanalysis courses.Itisacompulsorycourseforallmathematicsmajors.Itisalsostrongly recommendedforstudentsthatmajorincomputerscience,physics,datascience, financialanalysis,andotherareasthatrequirealotofanalyticalskills.Some standardtextbooksinmathematicalanalysisincludetheclassicalonebyApostol [Apo74]andRudin[Rud76],andthemodernonebyBartle[BS92],Fitzpatrick [Fit09],Abbott[Abb15],Tao[Tao16, Tao14]andZorich[Zor15, Zor16].

Thisbookisthefirstvolumeofthetextbooksintendedforaone-yearcoursein mathematicalanalysis.Weintroducethefundamentalconceptsinapedagogical way.Lotsofexamplesaregiventoillustratethetheories.Weassumethatstudents arefamiliarwiththematerialofcalculussuchasthoseinthebook[SCW20]. Thus,wedonotemphasizeonthecomputationtechniques.Emphasisisputon buildingupanalyticalskillsthroughrigorousreasonings.

Besidescalculus,itisalsoassumedthatstudentshavetakenintroductory coursesindiscretemathematicsandlinearalgebra,whichcoverstopicssuchas logic,sets,functions,vectorspaces,innerproducts,andquadraticforms.Whenever needed,theseconceptswouldbebrieflyrevised.

Inthisbook,wehavedefinedallthemathematicaltermsweusecarefully. Whilemostofthetermshavestandarddefinitions,someofthetermsmayhave definitionsdeferfromauthorstoauthors.Thereadersareadvisedtocheckthe definitionsofthetermsusedinthisbookwhentheyencounterthem.Thiscanbe easilydonebyusingthesearchfunctionprovidedbyanyPDFviewer.Thereaders arealsoencouragedtofullyutilizethehyper-referencingprovided.

TeoLeePeng

Chapter1

TheRealNumbers

1.1Logic,SetsandFunctions

Inthissection,wegiveabriefreviewofpropositionallogic,setsandfunctions. Itisassumedthatstudentshavetakenanintroductorycoursewhichcoversthese topics,suchasacourseindiscretemathematics[Ros18].

Definition1.1Proposition

A proposition,usuallydenotedby p,isadeclarativesentencethatiseither trueorfalse,butnotboth.

Definition1.2NegationofaProposition

If p isaproposition, ¬p isthe negation of p.Theproposition p istrueif andonlyifthenegation ¬p isfalse.

Fromtwopropositions p and q,wecanapplylogicaloperatorsandobtaina compoundproposition.

Definition1.3ConjunctionofPropositions

If p and q arepropositions, p ∧ q isthe conjunction of p and q,readas"p and q".Theproposition p ∧ q istrueifandonlyifboth p and q aretrue.

Definition1.4DisjunctionofPropositions

If p and q arepropositions, p ∨ q isthe disjunction of p and q,readas"p or q".Theproposition p ∨ q istrueifandonlyifeither p istrueor q istrue.

Definition1.5ImplicationofPropositions

If p and q arepropositions,theproposition p → q isreadas"p implies q". Itisfalseifandonlyif p istruebut q isfalse.

p → q canalsobereadas"if p then q or"p onlyif q".Inmathematics,we usuallywrite p =⇒ q insteadof p → q

Definition1.6DoubleImplication

If p and q arepropositions,theproposition p ←→ q isreadas"p ifand onlyif q".Itistheconjunctionof p → q and q → p.Hence,itistrueifand onlyifboth p and q aretrue,orboth p and q arefalse.

Thestament“p ifandonlyif q”isoftenexpressedas p ⇐⇒ q Twocompoundpropositions p and q aresaidtobelogicallyequivalent,denoted by p ≡ q,providedthat p istrueifandonlyif q istrue.

Logicalequivalencesareimportantforworkingwithmathematicalproofs. Someequivalencessuchascommutativelaw,associativelaw,distributivelaware obvious.Otherimportantequivalencesarelistedinthetheorembelow.

Theorem1.1LogicalEquivalences

Let p, q, r bepropositions.

1. p → q ≡¬p ∨ q

2. DeMorgan’sLaw

Averyimportantequivalenceistheequivalenceofanimplicationwithits contrapositive.

Theorem1.2Contraposition

If p and q arepropositions, p → q isequivalentto ¬q →¬p.

Chapter1.TheRealNumbers3

Inmathematics,weareoftendealingwithstatementsthatdependonvariables. Quantifiersareusedtospecifytheextenttowhichsuchastatementistrue.Two commonlyusedquantifiersare"forall"(∀)and"thereexists"(∃). Fornegationofstatementswithquantifiers,wehavethefollowinggeneralized DeMorgan’slaw.

Theorem1.3GeneralizedDeMorgan’sLaw

Fornestedquantifiers,theorderingisimportantifdifferenttypesofquantifiers areinvolved.Forexample,thestatement

x ∃yx + y =0 isnotequivalenttothestatement

y ∀xx + y =0

Whenthedomainsfor x and y areboththesetofrealnumbers,thefirststatement istrue,whilethesecondstatementisfalse.

Foraset A,weusethenotation x ∈ A todenote x isanelementoftheset A; andthenotation x/ ∈ A todenote x isnotanelementof A

Definition1.7EqualSets

Twosets A and B areequaliftheyhavethesameelements.Inlogical expression, A = B ifandonlyif

Definition1.8Subset

If A and B aresets,wesaythat A isa subset of B,denotedby A ⊂ B, ifeveryelementof A isanelementof B.Inlogicalexpression, A ⊂ B meansthat

x ∈ A =⇒ x ∈ B.

1. ¬ (∀xP (x)) ≡∃x ¬P (x)

2. ¬ (∃xP (x)) ≡∀x ¬P (x)

Chapter1.TheRealNumbers4

When A isasubsetof B,wewillalsosaythat A iscontainedin B,or B contains A.

Wesaythat A isa propersubset of B if A isasubsetof B and A = B.In sometextbooks,thesymbol"⊆"isusedtodenotesubset,andthesymbol"⊂" isreservedforpropersubset.Inthisbook,wewillnotmakesuchadistinction. Wheneverwewrite A ⊂ B,itmeans A isasubsetof B,notnecessaryaproper subset.

Thereareoperationsthatcanbedefinedonsets,suchasunion,intersection, differenceandcomplement.

Definition1.9UnionofSets

If A and B aresets,the union of A and B istheset A ∪ B whichcontains allelementsthatareeitherin A orin B.Inlogicalexpression,

Definition1.10IntersectionofSets

If A and B aresets,the intersection of A and B istheset A ∩ B which containsallelementsthatareinboth A and B.Inlogicalexpression,

Definition1.11DifferenceofSets

If A and B aresets,thedifferenceof A and B istheset A \ B which containsallelementsthatarein A andnotin B.Inlogicalexpression,

Definition1.12ComplementofaSet

If A isasetthatiscontainedinauniversalset U ,the complement of A in U istheset AC whichcontainsallelementsthatarein U butnotin A.In logicalexpression,

Chapter1.TheRealNumbers5

Sinceauniversalsetcanvaryfromcontexttocontext,wewillusuallyavoid usingthenotation AC anduse U \ A insteadforthecomplementof A in U .The advantageofusingthenotation AC isthatDeMorgan’slawtakesamoresuccint form.

Proposition1.4DeMorgan’sLawforSets

If A and B aresetsinauniversalset U ,and AC and BC aretheir complementsin U ,then

1. (A ∪ B)C = AC ∩ BC

2. (A ∩ B)C = AC ∪ BC

Definition1.13Functions

When A and B aresets,a function f from A to B,denotedby f : A → B, isacorrespondencethatassignseveryelementof A auniqueelementin B.

If a isin A,the image of a underthefunction f isdenotedby f (a),andit isanelementof B.

A iscalledthe domain of f ,and B iscalledthe codomain of f

Definition1.14ImageofaSet

If f : A → B isafunctionand C isasubsetof A,theimageof C under f istheset

f (C)= {f (c) | c ∈ C}

f (A) iscalledtherangeof f

Definition1.15PreimageofaSet

If f : A → B isafunctionand D isasubsetof B,thepreimageof D under f istheset

f 1(D)= {a ∈ A | f (a) ∈ D} .

Noticethat f 1(D) isanotation,itdoesnotmeanthatthefunction f hasan inverse.

Next,weturntodiscussinjectivityandsurjectivityoffunctions.

Chapter1.TheRealNumbers6

Definition1.16Injection

Wesaythatafunction f : A → B isan injection,orthefunction f : A → B is injective,orthefunction f : A → B is one-to-one,ifnopairof distinctelementsof A aremappedtothesameelementof B.Namely,

Usingcontrapositive,afunctionisinjectiveprovidedthat

Definition1.17Surjection

Wesaythatafunction f : A → B isa surjection,orthefunction f : A → B is surjective,orthefunction f : A → B is onto,ifeveryelementof B istheimageofsomeelementin A.Namely,

Equivalently, f : A → B issurjectiveiftherangeof f is B.Namely, f (A)= B

Definition1.18Bijection

Wesaythatafunction f : A → B isa bijection,orthefunction f : A → B isbijective,ifitisbothinjectiveandsurjective.

Abijectionisalsocalleda one-to-onecorrespondence. Finally,wewouldliketomakearemarkaboutsomenotations.If f : A → B isafunctionwithdomain A,and C isasubsetof A,therestrictionof f to C is thefunction f |C : C → B definedby f |C (c)= f (c) forall c ∈ C.Whenno confusionarises,wewilloftendenotethisfunctionsimplyas f : C → B

Chapter1.TheRealNumbers7

1.2TheSetofRealNumbersandItsSubsets

Inthissection,weintroducethesetofrealnumbersusinganintuitiveapproach.

Definition1.19NaturalNumbers

Thesetof naturalnumbers N isthesetthatcontainsthecountingnumbers, 1,2,3 ,whicharealsocalledpositiveintegers.

N isaninductiveset.Thenumber1isthesmallestelementofthisset.If n is anaturalnumber,then n +1 isalsoanaturalnumber. Thenumber0correspondstonothing.

Foreverypositiveinteger n, n isanumberwhichproduces0whenaddsto n.Thisnumber n iscalledthenegativeof n,ortheadditiveinverseof n 1, 2, 3, ...,arecallednegativeintegers.

Definition1.20Integers

Thesetof integers Z isthesetthatcontainsallpositiveintegers,negative integersand0.

Wewillalsousethenotation Z+ todenotethesetofpositiveintegers.

Definition1.21RationalNumbers

Thesetof rationalnumbers Q isthesetdefinedas

Eachrationalnumberisaquotientoftwointegers,wherethedenominatoris nonzero.Thesetofintegers Z isasubsetofthesetofrationalnumbers Q

Everyrationalnumber m/n hasadecimalexpansion.Forexample,

Thedecimalexpansionofarationalnumberiseitherfiniteorperiodic.

Chapter1.TheRealNumbers8

Definition1.22RealNumbers

Thesetof realnumbers R isintuitivelydefinedtobethesetthatcontains alldecimalnumbers,whichisnotnecessaryperiodic.

Thesetofrealnumberscontainsthesetofrationalnumbers Q asasubset.If arealnumberisnotarationalnumber,wecallitan irrationalnumber.Theset ofirrationalnumbersis R \ Q Ithasbeenlongknownthattherearerealnumbersthatarenotrationalnumbers. Thebestexampleisthenumber √2,whichappearsasthelengthofthediagonal ofaunitsquare(seeFigure 1.1).

√2.

Theadditionandmultiplicationoperationsdefinedonthesetofnaturalnumbers canbeextendedtothesetofrealnumbersconsistently.

If a and b arerealnumbers, a + b istheadditionof a and b,and ab isthe multiplicationof a and b.

If a and b arepositiverealnumbers, a+b and ab arealsopositiverealnumbers.

Thesetofrealnumberswiththeadditionandmultuplicationoperationsis afield,whichyouwilllearninabstractalgebra.Theseoperationssatisfythe followingproperties.

Figure1.1:Thenumber

PropertiesofRealNumbers

1. CommutativityofAddition

2. AssociativityofAddition

3. AdditiveIdentity a +0=0+

0iscalledtheadditiveidentity.

4. AdditiveInverse

Foreveryrealnumber a,thenegativeof a,denotedby a,satisfies

5. CommutativityofMultiplication

6. AssociativityofMultiplication

7. MultiplicativeIdentity

1iscalledthemultiplicativeidentity.

8. MultiplicativeInverse

Foreverynonzerorealnumber a,thereciprocalof a,denotedby 1/a, satisfies

9. Distributivity

Chapter1.TheRealNumbers10

Thesetofcomplexnumbers C isthesetthatcontainsallnumbersofthe form a + ib,where a and b arerealnumbers,and i isthepurelyimaginary numbersuchthat i2 = 1.Itcontainsthesetofrealnumbers R asasubset. Additionandmultiplicationcanbeextendedtothesetofcomplexnumbers.These twooperationsoncomplexnumbersalsosatisfyallthepropertieslistedabove. Nevertheless,weshallfocusonthesetofrealnumbersinthiscourse.

Therearespecialsubsetsofrealnumberswhicharecalled intervals.There areninetypesofintervals,fourtypesarefinite,fivetypesaresemi-infiniteor infinite.Theirdefinitionsareasfollows.

FiniteIntervals

1. (a,b)= {x ∈ R | a<x<b}

2. [a,b)= {x ∈ R | a ≤ x<b}

3. (a,b]= {x ∈ R | a<x ≤ b}

4. [a,b]= {x ∈ R | a ≤ x ≤ b}

Fortheintervals (a,b), [a,b), (a,b], [a,b],thepoints a and b arethe endpoints oftheinterval,whileanypoint x with a<x<b isan interiorpoint.

Semi-InfiniteorInfiniteIntervals

5. (a, ∞)= {x ∈ R | x>a}

6. [a, ∞)= {x ∈ R | x ≥ a}

7. (−∞,a)= {x ∈ R | x<a}

8. (−∞,a]= {x ∈ R | x ≤ a}

9. (−∞, ∞)= R.

Fortheintervals (a, ∞), [a, ∞), (−∞,a) and (−∞,a], a isthe endpoint of theinterval,whileanyotherpointsintheintervalbesides a isan interiorpoint

Thesetofnaturalnumbersisawell-orderedset.Everynonemptysubset ofpositiveintegershasasmallestelement.Thisstatementisequivalenttothe

Chapter1.TheRealNumbers11

principleofmathematicalinduction,whichisoneoftheimportantstrategiesin provingmathematicalstatements.

Proposition1.5PrincipleofMathematicalInduction

Let P (n) beasequenceofstatementsthatareindexedbythesetofpositive integers Z+.Assumethatthefollowingtwoassertionsaretrue.

1. Thestatement P (1) istrue.

2. Foreverypositiveinteger n,ifthestatement P (n) istrue,thestatement P (n +1) isalsotrue.

Thenwecanconcludethatforallpositiveintegers n,thestatement P (n) is true.

Beforeendingthissection,letusdiscusstheabsolutevalueandsomeuseful inequalities.

Definition1.23AbsoluteValue

Givenarealnumber x,the absolutevalue of x,denotedby |x|,isdefined tobethenonnegativenumber

Inparticular, |− x

Forexample, |

Theabsolutevalue |x| canbeinterpretedasthedistancebetweenthenumber x andthenumber 0 onthenumberline.Foranytworealnumbers x and y, |x y| isthedistancebetween x and y.Hence,theabsolutevaluecanbeusedtoexpress aninterval.

Chapter1.TheRealNumbers12

IntervalsDefinedbyAbsoluteValues

Let a bearealnumber.

1. If r isapositivenumber, |x a| <r ⇐⇒−r<x a<r ⇐⇒ x ∈ (a r,a + r).

2. If r isanonnegativenumber,

Absolutevaluesbehavewellwithrespecttomultiplicationoperation.

Proposition1.6

Givenrealnumbers x and y,

Ingeneral, |x + y| isnotequalto |x| + |y|.Instead,wehaveaninequality, knownasthetriangleinequality,whichisveryimportantinanalysis.

Proposition1.7TriangleInequality

Givenrealnumbers x and y,

Thisisprovedbydiscussingallfourpossiblecaseswhere x ≥ 0 or x< 0, y ≥ 0 or y< 0.

Acommonmistakestudentstendtomakeistoreplacebothplussignsinthe triangleequalitydirectlybyminussigns.Thisistotallyassurd.Thecorrectoneis

Fortheinequalityintheotherdirection,wehave

Chapter1.TheRealNumbers13

Proposition1.8

Givenrealnumbers x and y,

Proof

Since |x y|≥ 0,thestatementisequivalentto

Bytriangleinequality,

Hence,

Bytriangleinequalityagain,

Hence,

Thiscompletestheproof.

Example1.1

If |x 5|≤ 2,showthat 9 ≤ x 2 ≤ 49

Solution

|x 5|≤ 2 implies 3 ≤ x ≤ 7.Thismeansthat x ispositive.The inequality x ≥ 3 thenimpliesthat x2 ≥ 9,andtheinequality x ≤ 7 implies that x2 ≤ 49.Therefore, 9 ≤ x 2 ≤ 49.

Finally,wehavetheusefulCauchy’sinequality.

Chapter1.TheRealNumbers14

Proposition1.9Cauchy’sInequality

Thisisjustaconsequenceof

AnimmediateconsequenceofCauchy’sinequalityisthearithmeticmeangeometricmeaninequality.Foranynonnegativenumbers a and b,thegeometric meanof a and b is √ab,andthearithmeticmeanis

Proposition1.10

Chapter1.TheRealNumbers15

Exercises1.2

Question1

Useinductiontoshowthatforanypositiveinteger n, n! ≥ 2n 1

Question2:Bernoulli’sInequality

Giventhat a> 1,useinductiontoshowthat

forallpositiveinteger n.

Question3

Let n beapositiveinteger.If c

,c2,...,cn arenumbersthatlieinthe interval (0, 1),showthat

1.3BoundedSetsandtheCompletenessAxiom

Inthissection,wediscussapropertyofrealnumberscalledcompleteness.The setofrationalnumbersdoesnothavethisproperty. First,weintroducetheconceptofboundedness.

Definition1.24Boundedness

Let S beasubsetof R

1. Wesaythat S is boundedabove ifthereisanumber c suchthat x ≤ c forall x ∈ S.

Sucha c iscalledanupperboundof S

2. Wesaythat S is boundedbelow ifthereisanumber b suchthat

≥ b forall x ∈ S.

Sucha b iscalledalowerboundof S

3. Wesaythat S is bounded ifitisboundedaboveandboundedbelow.In thiscase,thereisanumber M suchthat

x|≤ M forall x ∈ S.

Letuslookatsomeexamples.

Example1.2

Determinewhethereachofthefollowingsetsofrealnumbersisbounded above,whetheritisboundedbelow,andwhetheritisbounded.

A = {x | x< 2}

B = {x | x> 2}

C = {x |− 2 <x< 2}.

Chapter1.TheRealNumbers17

Solution

(a) Theset A isboundedabovesinceeveryelementof A islessthanor equalto2.Itisnotboundedbelow,andsoitisnotbounded.

(b) Theset B isboundedbelowsinceeveryelementof B islargerthanor equalto 2.Itisnotboundedabove,andsoitisnotbounded.

Figure1.2:Thesets A, B, C inExample 1.2

If S isasetofrealnumbers,the negative of S,denotedby S,istheset S = {−x | x ∈ S}

Forexample,theset B = {x | x> 2} isthenegativeoftheset A = {x | x< 2}, theset C = {x |− 2 <x< 2} isthenegativeofitself(seeFigure 1.2).Itis obviousthat S isboundedaboveifandonlyif S isboundedbelow. Next,werecallthedefinitionofmaximumandminimumofaset.

Chapter1.TheRealNumbers18

Definition1.25MaximumandMinimum

Let S beanonemptysubsetofrealnumbers.

1. Anumber c iscalledthe largestelement or maximum of S if c isan elementof S and x ≤ c forall x ∈ S.

Ifthemaximumoftheset S exists,wedenoteisby max S.

2. Anumber b iscalledthe smallestelement or minimum of S if b isan elementof S and x ≥ b forall x ∈ S.

Iftheminimumoftheset S exists,wedenoteitby min S.

Obviously, b isthemaximumofaset S ifandonlyif b istheminimumof theset S.

Example1.3

Fortheset S1 =[ 2, 2], 2 istheminimum,and 2 isthemaximum.

Fortheset S2 =[ 2, 2), 2 istheminimum,andthereisnomaximum.

Thisexampleshowsthataboundedsetdoesnotnecessarilyhavemaximum orminimum.However,afinitesetalwayshaveamaximumandaminimum.

Proposition1.11

If S isafiniteset,then S hasamaximumandaminimum.

Next,weintroducetheconceptofleastupperbound.

Chapter1.TheRealNumbers19

Definition1.26LeastUpperBound

Let S beanonemptysubsetofrealnumbersthatisboundedabove,and let US bethesetofupperboundsof S.Then US isanonemptysetthatis boundedbelow.If US hasasmallestelement u,wesaythat u isthe least upperbound or supremum of S,anddenoteitby u =sup S.

Example1.4

Forthesets S1 =[ 2, 2] and S2 =[ 2, 2), sup S1 =sup S2 =2

Noticethat sup S,ifexists,isnotnecessaryanelementof S.Thefollowing propositiondepictstherelationbetweenthemaximumofaset(ifexists)andits leastupperbound.

Proposition1.12SupremumandMaximum

Let S beanonemptysubsetofrealnumbers.Then S hasamaximumifand onlyif S isboundedaboveand sup S isin S

Onenaturalquestiontoaskis,if S isanonemptysubsetofrealnumbersthat isboundedabove,does S necessarilyhavealeastupperbound.Thecompleteness axiomassertsthatthisistrue.

CompletenessAxiom

If S isanonemptysubsetofrealnumbersthatisboundedabove,then S hasaleastupperbound.

Thereasonthisisformulatedasanaxiomiswecannotprovethisfromour intuitivedefinitionofrealnumbers.Therefore,wewillassumethisasafactfor thesetofrealnumbers.Alotsoftheoremsthatwearegoingtoderivelaterisa consequenceofthisaxiom.

Actually,thesetofrealnumberscanbeconstructedaxiomatically,takenit

Chapter1.TheRealNumbers20

tobeasetthatcontainsthesetofrationalnumbers,satisfyingallproperties ofadditionandmultiplicationoperations,aswellasthecompletenessaxiom. However,thisisatediousconstructionandwilldriftustoofar. Toshowthatthecompletenessaxiomisnotcompletelytrivial,weshowin Example 1.6 thatifweonlyconsiderthesetofrationalnumbers,wecanfinda subsetofrationalnumbers A thatisboundedabovebutdoesnothavealeastupper boundinthesetofrationalnumbers.Welookatthefollowingexamplefirst.

Example1.5

Definethesetofrealnumbers S by

= x ∈ R | x 2 < 2

Showthat S isnonemptyandisboundedabove.Concludethattheset A = x ∈ Q | x 2 < 2 isalsononemptyandisboundedabovebyarationalnumber.

Solution

Thenumber1isin S,andso S isnonempty.Forany x ∈ S, x2 < 2 < 4, andhence x< 2.Thisshowsthat S isboundedaboveby2.Since1and2 arerationalnumbers,thesamereasoningshowsthattheset A isnonempty andisboundedabovebyarationalnumber.

Example1.6

Considertheset

= x ∈ Q | x 2 < 2

ByExample 1.5, A isanonemptysubsetofrationalnumbersthatis boundedaboveby2.Let UA bethesetofupperboundsof A in Q.Namely,

Showthat UA doesnothaveasmallestelement.

Chapter1.TheRealNumbers21

Solution

Weuseproofbycontradiction.Assumethat UA hasasmallestelement c1, whichisanupperboundof A thatissmallerthanorequaltoanyupper boundof A.Thenforany x ∈ A, x 2 ≤ c1.

Since 1 isin A, c1 isapositiverationalnumber.Hence,therearepoitive integers p and q suchthat c1 = p q .

Sincetherearenorationalnumberswhosesquareis2,wemusthaveeither c2 1 < 2 or c2 1 > 2

Definethepositiverationalnumber c2 by

Noticethat

and

Case1: c2 1 < 2

Inthiscase, p2 < 2q2.Itfollowsthat c1 <c2 and c2 2 < 2.Butthen c1 and c2 arebothin A,and c2 isanelementin A thatislargerthan c1,which contradictsto c1 isanupperboundof A.Hence,wecannothave c2 1 < 2.

Chapter1.TheRealNumbers22

Case2: c2 1 > 2.

Inthiscase, p2 > 2q2.Itfollowsthat c1 >c2 and c2 2 > 2.Since c2 2 > 2,we findthatforany x ∈ A,

Thus,

Inparticular, c2 isalsoanupperboundof A.Namely, c2 isin UA.Butthen c1 and c2 arebothin UA and c1 >c2.Thiscontradictsto c1 isthesmallest elementin UA.Hence,wecannothave c2 1 > 2

SincebothCase1andCase2leadtocontradictions,weconcludethat UA doesnothaveasmallestelement.

Inthesolutionabove,theconstructionofthepositiverationalnumber c2 seems abitadhoc.Infact,wecandefine c2 by c2 = mp +2nq np + mq foranypositiveintegers m and n with m2 > 2n2.Thentheproofstillworks.

Nowletusseehowcompletenessaxiomisusedtoguaranteethatthereisa realnumberwhosesquareis2.

Example1.7

Usecompletenessaxiomtoshowthatthereisapositiverealnumber c such that

Definethesetofrealnumbers S by S

Example 1.5 assertsthat S isanonemptysubsetofrealnumbersthatis boundedabove.Completenessaxiomassertsthat S hasaleastupperbound c.

Chapter1.TheRealNumbers23

Since 1 isin S, c ≥ 1.Wearegoingtoprovethat c2 =2 usingproofby contradiction.If c2 =2,then c2 < 2 or c2 > 2.

Case1: c2 < 2

Let d =2 c2.Then 0 <d ≤ 1.Definethenumber c1 by

1 = c + d 4c .

Then c1 >c,and

Thisimpliesthat c1 isanelementof S thatislargerthan c,whichcontradicts to c isanupperboundof S

Case2: c2 > 2.

Let d = c2 2.Then d> 0.Definethenumber c1 by c1 = c d 2c

Then c1 <c,and

Thisimpliesthat c1 isanupperboundof S thatissmallerthan c,which contradictsto c istheleastupperboundof S. Sinceweobtainacontradictionif c2 =2,wemusthave c2 =2.

Infact,thecompletenessaxiomcanbeusedtoshowthatforanypositivereal number a,thereisapositiverealnumber c suchthat c 2 = a.

Wedenotethisnumber c as √a,calledthepositivesquarerootof a.Thenumber b = √a isanotherrealnumbersuchthat b2 = a

Moregenerally,if n isapositiveinteger, a isapositiverealnumber,thenthere isapositiverealnumber c suchthat cn = a.Wedenotethisnumber c by

c = n √a,

Chapter1.TheRealNumbers24

calledthepositive nth-rootof a Usingtheinterplaybetweenasetanditsnegative,wecandefinethegreatest lowerboundofasetthatisboundedbelow.

Definition1.27GreatestLowerBound

Let S beanonemptysubsetofrealnumbersthatisboundedbelow,and let LS bethesetoflowerboundsof S.Then LS isanonemptysetthatis boundedabove.If LS hasalargestelement ℓ,wesaythat ℓ isthe greatest lowerbound or infimum of S,anddenoteitby

=inf S.

Fromthecompletenessaxiom,wehavethefollowing.

Theorem1.13

If S isanonemptysubsetofrealnumbersthatisboundedbelow,then S hasagreatestlowerbound.

Foranonemptyset S thatisbounded,ithasaleastupperbound sup S anda greatestlowerbound inf S.Thefollowingisquiteobvious.

Proposition1.14

If S isaboundednonemptysubsetofrealnumbers,ithasaleastupper bound sup S andagreatestlowerbound inf S.Moreover, inf S ≤ sup S, and inf S =sup S ifandonlyif S containsexactlyoneelement.

Letusemphasizeagainthecharacterizationoftheleastupperboundand greatestlowerboundofaset.