INTRODUCTION TO REAL ANALYSIS

586 Pages·2013·3.55 MB·English

INTRODUCTION
TO REAL ANALYSIS
William F. Trench
AndrewG. Cowles DistinguishedProfessorEmeritus
DepartmentofMathematics
TrinityUniversity
SanAntonio,Texas, USA
[email protected]
This book has been judged to meet the evaluation criteria set by
the Editorial Board of the American Institute of Mathematics in
connection with the Institute’s Open TextbookInitiative. It may
becopied,modified,redistributed,translated,andbuiltuponsub-
jecttothe CreativeCommons
Attribution-NonCommercial-ShareAlike3.0 UnportedLicense.
FREE DOWNLOADABLE SUPPLEMENTS
FUNCTIONS DEFINED BY IMPROPER INTEGRALS
THE METHOD OF LAGRANGE MULTIPLIERS
LibraryofCongressCataloging-in-PublicationData
Trench,WilliamF.
Introductiontorealanalysis / WilliamF.Trench
p.cm.
ISBN0-13-045786-8
1.MathematicalAnalysis. I. Title.
QA300.T6672003
515-dc21 2002032369
FreeHyperlinkedEdition2.04December2013
ThisbookwaspublishedpreviouslybyPearsonEducation.
Thisfreeeditionismade availableinthehopethatitwillbeusefulasatextbookorrefer-
ence. Reproductionispermittedforany validnoncommercialeducational, mathematical,
or scientific purpose. However, charges for profit beyond reasonable printing costs are
prohibited.
Acompleteinstructor’[email protected],sub-
ject to verification of the requestor’s faculty status. Although this book is subject to a
Creative Commons license, thesolutionsmanual is not. The authorreserves allrightsto
themanual.
TO BEVERLY
Contents
Preface vi
Chapter 1 The Real Numbers 1
1.1 The Real Number System 1
1.2 Mathematical Induction 10
1.3 The Real Line 19
Chapter 2 Differential Calculus of Functions of One Variable 30
2.1 Functions and Limits 30
2.2 Continuity 53
2.3 Differentiable Functions of One Variable 73
2.4 L’Hospital’s Rule 88
2.5 Taylor’s Theorem 98
Chapter 3 Integral Calculus of Functions of One Variable 113
3.1 Definition of the Integral 113
3.2 Existence of the Integral 128
3.3 Properties of the Integral 135
3.4 Improper Integrals 151
3.5 A More Advanced Look at the Existence
of the Proper Riemann Integral 171
Chapter 4 Infinite Sequences and Series 178
4.1 Sequences of Real Numbers 179
4.2 Earlier Topics Revisited With Sequences 195
4.3 Infinite Series of Constants 200
iv
Contents v
4.4 Sequences and Series of Functions 234
4.5 Power Series 257
Chapter 5 Real-Valued Functions of Several Variables 281
5.1 Structure of RRRn 281
5.2 Continuous Real-Valued Function of n Variables 302
5.3 Partial Derivatives and the Differential 316
5.4 The Chain Rule and Taylor’s Theorem 339
Chapter 6 Vector-Valued Functions of Several Variables 361
6.1 Linear Transformations and Matrices 361
6.2 Continuity and Differentiability of Transformations 378
6.3 The Inverse Function Theorem 394
6.4. The Implicit Function Theorem 417
Chapter 7 Integrals of Functions of Several Variables 435
7.1 Definition and Existence of the Multiple Integral 435
7.2 Iterated Integrals and Multiple Integrals 462
7.3 Change of Variables in Multiple Integrals 484
Chapter 8 Metric Spaces 518
8.1 Introduction to Metric Spaces 518
8.2 Compact Sets in a Metric Space 535
8.3 Continuous Functions on Metric Spaces 543
Answers to Selected Exercises 549
Index 563
Preface
Thisisatextforatwo-termcourseinintroductoryrealanalysisforjuniororseniormath-
ematics majors and science students with a serious interest in mathematics. Prospective
educators ormathematicallygiftedhighschoolstudentscan also benefitfromthe mathe-
maticalmaturitythatcanbegainedfromanintroductoryrealanalysiscourse.
Thebookisdesignedtofillthegapsleftinthedevelopmentofcalculusasitisusually
presentedinanelementarycourse,andtoprovidethebackgroundrequiredforinsightinto
moreadvanced coursesinpureandappliedmathematics. Thestandardelementary calcu-
lussequenceistheonlyspecificprerequisiteforChapters1–5,whichdealwithreal-valued
functions.(However,otheranalysisorientedcourses,suchaselementarydifferentialequa-
tion, also provide useful preparatory experience.) Chapters 6 and 7 require a working
knowledgeofdeterminants,matricesandlineartransformations,typicallyavailablefroma
firstcourseinlinearalgebra. Chapter8isaccessibleaftercompletionofChapters1–5.
Withouttakinga positionfororagainstthecurrentreforms inmathematics teaching, I
thinkit is fair to say that the transitionfrom elementary courses such as calculus, linear
algebra, and differential equations to a rigorous real analysis course is a bigger step to-
day than itwas justa few years ago. To make this step today’sstudentsneed more help
thantheirpredecessors did,andmustbecoached andencouragedmore. Therefore, while
strivingthroughouttomaintainahighlevelofrigor,Ihavetriedtowriteasclearlyandin-
formallyaspossible.InthisconnectionIfinditusefultoaddressthestudentinthesecond
person. I have included295completely workedoutexamples to illustrateand clarifyall
majortheoremsanddefinitions.
I have emphasized careful statements of definitionsand theorems and have triedto be
complete anddetailedinproofs, except foromissionslefttoexercises. I givea thorough
treatmentofreal-valuedfunctionsbefore consideringvector-valuedfunctions. Inmaking
thetransitionfromonetoseveralvariablesandfromreal-valuedtovector-valuedfunctions,
Ihave lefttothestudentsomeproofsthatareessentiallyrepetitionsofearliertheorems. I
believethatworkingthroughthedetailsofstraightforwardgeneralizationsofmoreelemen-
taryresultsisgoodpracticeforthestudent.
Great care has gone into the preparation of the 761 numbered exercises, many with
multipleparts. Theyrangefromroutinetoverydifficult. Hintsare providedforthemore
difficultpartsoftheexercises.
vi
Preface vii
Organization
Chapter1isconcerned withthe realnumber system. Section1.1 beginswitha briefdis-
cussionoftheaxiomsforacompleteorderedfield, butnoattemptismade todevelopthe
realsfromthem;rather,itisassumedthatthestudentisfamiliarwiththeconsequencesof
these axioms,except forone: completeness. Since thedifferencebetween arigorousand
nonrigoroustreatment of calculus can be described largely in terms of the attitude taken
toward completeness, I have devoted considerable effort to developingits consequences.
Section 1.2 is about induction. Although this may seem out of place in a real analysis
course, I have foundthatthe typicalbeginningreal analysis studentsimplycannotdo an
inductionproofwithoutreviewingthemethod. Section1.3isdevotedtoelementarysetthe-
oryandthetopologyoftherealline,endingwiththeHeine-BorelandBolzano-Weierstrass
theorems.
Chapter2covers thedifferentialcalculus offunctionsofone variable: limits, continu-
ity,differentiablility,L’Hospital’srule,andTaylor’stheorem. Theemphasisisonrigorous
presentation of principles; no attempt is made to develop the properties of specific ele-
mentary functions. Even though this may not be done rigorouslyin most contemporary
calculus courses, I believe thatthestudent’stime is betterspentonprinciplesrather than
onreestablishingfamiliarformulasandrelationships.
Chapter 3 is to devoted to the Riemann integral of functionsof one variable. In Sec-
tion3.1theintegralisdefinedinthestandardwayintermsofRiemannsums. Upperand
lowerintegralsarealsodefinedthereandusedinSection3.2tostudytheexistenceofthe
integral. Section3.3isdevotedtopropertiesoftheintegral. Improperintegralsarestudied
inSection3.4. Ibelieve thatmytreatment ofimproperintegrals ismore detailedthan in
mostcomparabletextbooks.AmoreadvancedlookattheexistenceoftheproperRiemann
integralisgiveninSection3.5,whichconcludeswithLebesgue’sexistencecriterion.This
section can be omitted withoutcompromising the student’s preparedness for subsequent
sections.
Chapter 4 treats sequences and series. Sequences of constant are discussed in Sec-
tion 4.1. I have chosen to make the concepts of limit inferior and limit superior parts
of this development, mainly because this permits greater flexibility and generality, with
littleextra effort, inthe studyof infiniteseries. Section 4.2 providesa briefintroduction
tothewayinwhichcontinuityanddifferentiabilitycanbestudiedbymeansofsequences.
Sections4.3–4.5treatinfiniteseriesofconstant,sequencesandinfiniteseriesoffunctions,
andpowerseries, againingreaterdetailthaninmostcomparable textbooks. Theinstruc-
torwho chooses nottocover these sections completely can omit the less standard topics
withoutlossinsubsequentsections.
Chapter5isdevotedtoreal-valuedfunctionsofseveral variables. Itbeginswitha dis-
cussionofthetoplogyofRn inSection5.1. Continuityanddifferentiabilityarediscussed
inSections5.2and5.3. ThechainruleandTaylor’stheoremarediscussedinSection5.4.
viii Preface
Chapter6coversthedifferentialcalculusofvector-valuedfunctionsofseveralvariables.
Section6.1reviewsmatrices, determinants,andlineartransformations,whichareintegral
parts of the differential calculus as presented here. In Section 6.2 the differential of a
vector-valuedfunctionisdefinedasalineartransformation,andthechainruleisdiscussed
intermsofcompositionofsuchfunctions. The inversefunctiontheoremisthesubjectof
Section6.3,where thenotionofbranchesofaninverse isintroduced. InSection6.4. the
implicitfunctiontheoremismotivatedbyfirstconsideringlineartransformationsandthen
statedandprovedingeneral.
Chapter7coverstheintegralcalculusofreal-valuedfunctionsofseveralvariables. Mul-
tiple integrals are defined in Section 7.1, first over rectangular parallelepipeds and then
overmoregeneralsets. Thediscussiondealswiththemultipleintegralofafunctionwhose
discontinuitiesformasetofJordancontentzero. Section7.2dealswiththeevaluationby
iteratedintegrals. Section7.3beginswiththedefinitionofJordanmeasurability,followed
byaderivationoftheruleforchangeofcontentunderalineartransformation,anintuitive
formulationofthe rule forchange of variables inmultipleintegrals, andfinallya careful
statementandproofoftherule. Theproofiscomplicated,butthisisunavoidable.
Chapter 8 deals with metric spaces. The concept and propertiesof a metric space are
introducedinSection8.1. Section8.2discusses compactness ina metric space, andSec-
tion8.3discussescontinuousfunctionsonmetricspaces.
Corrections–mathematicalandtypographical–arewelcomeandwillbeincorporatedwhen
received.
WilliamF.Trench
[email protected]
Home: 659HopkintonRoad
Hopkinton,NH03229
CHAPTER 1
The Real Numbers
INTHISCHAPTERwebeginthestudyoftherealnumbersystem. Theconceptsdiscussed
herewillbeusedthroughoutthebook.
SECTION 1.1 deals with the axioms that define the real numbers, definitions based on
them,andsomebasicpropertiesthatfollowfromthem.
SECTION1.2emphasizestheprincipleofmathematicalinduction.
SECTION 1.3 introduces basic ideas of set theory in the context of sets of real num-
bers. Inthissectionweprovetwofundamentaltheorems: theHeine–BorelandBolzano–
Weierstrasstheorems.
1.1 THE REAL NUMBER SYSTEM
Havingtakencalculus,youknowalotabouttherealnumbersystem; however, youprob-
ably do not knowthat all its propertiesfollowfrom a few basic ones. Althoughwe will
notcarryoutthedevelopmentoftherealnumbersystemfromthesebasicproperties,itis
usefulto state them as a startingpointforthe studyofreal analysisand also to focuson
oneproperty,completeness,thatisprobablynewtoyou.
Field Properties
The real number system (which we will often call simply the reals) is first of all a set
a;b;c;::: on which the operations of addition and multiplication are defined so that
f g
every pair of real numbers has a unique sum and product, both real numbers, with the
followingproperties.
(A) a b b aandab ba(commutativelaws).
C D C D
(B) .a b/ c a .b c/and.ab/c a.bc/(associativelaws).
C C D C C D
(C) a.b c/ ab ac (distributivelaw).
C D C
(D) Therearedistinctrealnumbers0and1suchthata 0 aanda1 afor alla.
C D D
(E) Foreachathereisarealnumber asuchthata . a/ 0,andifa 0, thereis
(cid:0) C (cid:0) D ¤
arealnumber1=asuchthata.1=a/ 1.
D
1
2 Chapter1TheRealNumbers
Themanipulativepropertiesoftherealnumbers,suchastherelations
.a b/2 a2 2ab b2;
C D C C
.3a 2b/.4c 2d/ 12ac 6ad 8bc 4bd;
C C D C C C
. a/ . 1/a; a. b/ . a/b ab;
(cid:0) D (cid:0) (cid:0) D (cid:0) D(cid:0)
and
a c ad bc
C .b;d 0/;
b C d D bd ¤
allfollowfrom(A)–(E).Weassumethatyouarefamiliarwiththeseproperties.
Asetonwhichtwooperationsaredefinedsoastohaveproperties(A)–(E)iscalleda
field. Therealnumbersystemisbynomeanstheonlyfield. Therationalnumbers(which
aretherealnumbersthatcanbewrittenasr p=q,wherepandqareintegersandq 0)
D ¤
alsoformafieldunderadditionandmultiplication.Thesimplestpossiblefieldconsistsof
twoelements,whichwedenoteby0and1,withadditiondefinedby
0 0 1 1 0; 1 0 0 1 1; (1.1.1)
C D C D C D C D
andmultiplicationdefinedby
0 0 0 1 1 0 0; 1 1 1 (1.1.2)
(cid:1) D (cid:1) D (cid:1) D (cid:1) D
(Exercise1.1.2).
The Order Relation
Therealnumbersystemisorderedbytherelation<,whichhasthefollowingproperties.
(F) Foreachpairofrealnumbersaandb,exactlyoneofthefollowingistrue:
a b; a <b; or b <a:
D
(G) Ifa <bandb <c,thena<c. (Therelation<istransitive.)
(H) Ifa <b,thena c <b cforanyc,andif0<c,thenac <bc.
C C
A field withan order relationsatisfying(F)–(H) is an ordered field. Thus, the real
numbersforman orderedfield. Therationalnumbersalsoformanorderedfield, butitis
impossibletodefineanorderonthefieldwithtwoelementsdefinedby(1.1.1)and(1.1.2)
soastomakeitintoanorderedfield(Exercise1.1.2).
Weassume thatyouare familiarwithotherstandardnotationconnectedwiththeorder
relation: thus,a > b meansthatb < a;a b meansthateithera b ora > b;a b
(cid:21) D (cid:20)
means that either a b or a < b; the absolute value of a, denoted by a , equals a if
D j j
a 0or aifa 0. (Sometimeswecall a themagnitudeofa.)
(cid:21) (cid:0) (cid:20) j j
Youprobablyknowthefollowingtheorem fromcalculus, butwe includethe prooffor
yourconvenience.