Basic Elements of Real Analysis

287 Pages·1998·1.06 MB·English

by  Murray H. Protter

MurrayH.Protter
DepartmentofMathematics
UniversityofCalifornia
Berkeley,CA94720
USA
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment Departmentof
SanFranciscoState EastHall Mathematics
University UniversityofMichigan UniversityofCalifornia
SanFrancisco,CA94132 AnnArbor,MI48109 atBerkeley
USA USA Berkeley,CA94720-3840
USA
Frontcoverillustration:f convergestof,butf1f doesnotconvergetof1f.(Seep.167of
n 0 n 0
textforexplanation.)
MathematicsSubjectClassification(1991):26-01,26-06,26A54
LibraryofCongressCataloging-in-PublicationData
Protter,MurrayH.
Basicelementsofrealanalysis/MurrayH.Protter.
p. cm.—(Undergraduatetextsinmathematics)
Includesbibliographicalreferencesandindex.
ISBN0-387-98479-8(hardcover:alk.paper)
1.Mathematicalanalysis. I.Title. II.Series.
QA300.P9678 1998
515—dc21 98-16913
(cid:2)c 1998Springer-VerlagNewYork,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout
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ISBN0-387-98479-8 Springer-Verlag NewYork Berlin Heidelberg SPIN10668224
To Barbara and Philip
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........................................... Preface
SometimeagoCharlesB.MorreyandIwroteAFirstCourseinRealAnaly-
sis,abookthatprovidesmaterialsufficientforacomprehensiveone-year
courseinanalysisforthosestudentswhohavecompletedastandardele-
mentarycourseincalculus.Thebookhasbeenthroughtwoeditions,the
secondofwhichappearedin1991;smallchangesandcorrectionsofmis-
printshavebeenmadeinthefifthprintingofthesecondedition,which
appearedrecently.
However, for many students of mathematics and for those students
whointendtostudyanyofthephysicalsciencesandcomputerscience,
the need is for a short one-semester course in real analysis rather than
a lengthy, detailed, comprehensive treatment. To fill this need the book
Basic Elements of Real Analysis provides, in a brief and elementary way,
themostimportanttopicsinthesubject.
Thefirstchapter,whichdealswiththerealnumbersystem,givesthe
reader the opportunity to develop facility in proving elementary theo-
rems.Sincemoststudentswhotakethiscoursehavespenttheireffortsin
developingmanipulativeskills,suchanintroductionpresentsawelcome
change.Thelastsectionofthischapter,whichestablishesthetechnique
of mathematical induction, is especially helpful for those who have not
previouslybeenexposedtothisimportanttopic.
Chapters2through5coverthetheoryofelementarycalculus,includ-
ingdifferentiationandintegration.Manyofthetheoremsthatare“stated
withoutproof”inelementarycalculusareprovedhere.
ItisimportanttonotethatboththeDarbouxintegralandtheRiemann
integral are described thoroughly in Chapter 5 of this volume. Here we
vii
viii Preface
establishtheequivalenceoftheseintegrals,thusgivingthereaderinsight
intowhatintegrationisallabout.
For topics beyond calculus, the concept of a metric space is crucial.
Chapter 6 describes topology in metric spaces as well as the notion of
compactness,especiallywithregardtotheHeine–Boreltheorem.
The subject of metric spaces leads in a natural way to the calculus
of functions in N-dimensional spaces with N >2. Here derivatives of
functions of N variables are developed, and the Darboux and Riemann
integrals, as described in Chapter 5, are extended in Chapter 7 to N-
dimensionalspace.
Infinite series is the subject of Chapter 8. After a review of the usual
tests for convergence and divergence of series, the emphasis shifts to
uniform convergence. The reader must master this concept in order to
understand the underlying ideas of both power series and Fourier se-
ries. Although Fourier series are not included in this text, the reader
should find it fairly easy reading once he or she masters uniform con-
vergence. For those interested in studying computer science, not only
FourierseriesbutalsotheapplicationofFourierseriestowavelettheoryis
recommended.(See,e.g.,TenLecturesonWaveletsbyIngridDaubechies.)
Therearemanyimportantfunctionsthataredefinedbyintegrals,the
integration taken over a finite interval, a half-infinite integral, or one
from −∞ to +∞. An example is the well-known Gamma function. In
Chapter9wedevelopthenecessarytechniquesfordifferentiationunder
theintegralsignofsuchfunctions(theLeibnizrule).Althoughdesirable,
thischapterisoptional,sincetheresultsarenotusedlaterinthetext.
Chapter10treatstheRiemann–Stieltjesintegral.Afteranintroduction
to functions of bounded variation, we define the R-S integral and show
how the usual integration-by-parts formula is a special case of this inte-
gral.ThegeneralityoftheRiemann–Stieltjesintegralisfurtherillustrated
by the fact that an infinite series can always be considered as a special
caseofaRiemann–Stieltjesintegral.
A subject that is heavily used in both pure and applied mathematics
is the Lagrange multiplier rule. In most cases this rule is stated without
proof but with applications discussed. However, we establish the rule
in Chapter 11 (Theorem 11.4) after developing the facts on the implicit
functiontheoremneededfortheproof.
Inthetwelfth,andlast,chapterwediscussvectorfunctionsinRN.We
prove the theorems of Green and Stokes and the divergence theorem,
not in full generality but of sufficient scope for most applications. The
ambitiousreadercangetamoregeneralinsighteitherbyreferringtothe
bookAFirstCourseinRealAnalysisorthetextPrinciplesofMathematical
AnalysisbyWalterRudin.
MurrayH.Protter
Berkeley,CA
........................................... Contents
Preface............................................................. vii
Chapter 1 The Real Number System ........................... 1
1.1AxiomsforaField.......................................... 1
1.2NaturalNumbersandSequences........................... 7
1.3Inequalities................................................ 11
1.4MathematicalInduction.................................... 19
Chapter 2 Continuity and Limits............................... 23
2.1Continuity................................................. 23
2.2Limits...................................................... 28
2.3One-SidedLimits........................................... 33
2.4LimitsatInfinity;InfiniteLimits........................... 37
2.5LimitsofSequences........................................ 42
Chapter 3 Basic Properties of Functions on R1................ 45
3.1TheIntermediate-ValueTheorem.......................... 45
3.2LeastUpperBound;GreatestLowerBound................. 48
3.3TheBolzano–WeierstrassTheorem......................... 54
3.4TheBoundednessandExtreme-ValueTheorems ........... 56
3.5UniformContinuity........................................ 57
3.6TheCauchyCriterion...................................... 61
3.7TheHeine–BorelTheorem................................. 63
ix