Real Analysis: With Proof Strategies

282 Pages·2020·11.127 MB·English

by  Daniel W. Cunningham

Real Analysis
Textbooks in Mathematics
Series editors:
Al Boggess, Kenneth H. Rosen
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https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
Real Analysis
With Proof Strategies
Daniel W. Cunningham
First edition published 2021
by CRC Press
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and by CRC Press
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© 2021 Daniel W. Cunningham
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Library of Congress Cataloging-in-Publication Data
Names: Cunningham, Daniel W., author.
Title: Real analysis : with proof strategies / Daniel W. Cunningham.
Description: First edition. | Boca Raton : Chapman & Hall, CRC Press, 2021.
| Series: Textbooks in mathematics | Includes bibliographical references
and index.
Identifiers: LCCN 2020034986 (print) | LCCN 2020034987 (ebook) | ISBN
9780367549657 (hardback) | ISBN 9781003091363 (ebook)
Subjects: LCSH: Mathematical analysis--Textbooks. | Functions of real
variables--Textbooks.
Classification: LCC QA300 .C86 2021 (print) | LCC QA300 (ebook) | DDC
515/.8--dc23
LC record available at https://lccn.loc.gov/2020034986
LC ebook record available at https://lccn.loc.gov/2020034987
ISBN: 9780367549657 (hbk)
ISBN: 9781003091363(ebk)
Typeset in Computer Modern font
by KnowledgeWorks Global Ltd.
Contents
Preface ix
Chapter 1(cid:4) Proofs, Sets, Functions, and Induction 1
1.1 PROOFS 1
1.1.1 Important Sets in Mathematics 1
1.1.2 How to Prove an Equation 3
1.1.3 How to Prove an Inequality 4
1.1.4 Important Properties of Absolute Value 6
1.1.5 Proof Diagrams 7
1.2 SETS 10
1.2.1 Basic Definitions of Set Theory 10
1.2.2 Set Operations 10
1.2.3 Indexed Families of Sets 11
1.2.4 Generalized Unions and Intersections 12
1.2.5 Unindexed Families of Sets 14
1.3 FUNCTIONS 15
1.3.1 Real-Valued Functions 16
1.3.2 Injections and Surjections 16
1.3.3 Composition of Functions 17
1.3.4 Inverse Functions 18
1.3.5 Functions Acting on Sets 19
1.4 MATHEMATICALINDUCTION 23
1.4.1 The Well-Ordering Principle 23
1.4.2 Proof by Mathematical Induction 23
Chapter 2(cid:4) The Real Numbers 29
2.1 INTRODUCTION 29
2.2 RISANORDEREDFIELD 30
v
vi (cid:4) Contents
2.2.1 The Absolute Value Function 33
2.3 THECOMPLETENESSAXIOM 36
2.3.1 Proofs on the Supremum of a Set 39
2.3.2 Proofs on the Infimum of a Set 40
2.3.3 Alternative Proof Strategies 43
2.4 THEARCHIMEDEANPROPERTY 47
2.4.1 The Density of the Rational Numbers 48
2.5 NESTEDINTERVALSTHEOREM 50
2.5.1 R is Uncountable 51
Chapter 3(cid:4) Sequences 55
3.1 CONVERGENCE 56
3.2 LIMITTHEOREMSFORSEQUENCES 68
3.2.1 Algebraic Limit Theorems 68
3.2.2 The Squeeze Theorem 72
3.2.3 Order Limit Theorems 73
3.3 SUBSEQUENCES 75
3.4 MONOTONESEQUENCES 79
3.4.1 The Monotone Subsequence Theorem 83
3.5 BOLZANO–WEIERSTRASSTHEOREMS 85
3.6 CAUCHYSEQUENCES 87
3.7 INFINITELIMITS 91
3.8 LIMITSUPERIORANDLIMITINFERIOR 93
3.8.1 The Limit Superior of a Bounded Sequence 93
3.8.2 The Limit Inferior of a Bounded Sequence 96
3.8.3 Connections and Relations 97
Chapter 4(cid:4) Continuity 101
4.1 CONTINUOUSFUNCTIONS 101
4.2 CONTINUITYANDSEQUENCES 110
4.3 LIMITSOFFUNCTIONS 112
4.4 CONSEQUENCESOFCONTINUITY 119
4.4.1 The Extreme Value Theorem 119
4.4.2 The Intermediate Value Theorem 120
4.5 UNIFORMCONTINUITY 124
Contents (cid:4) vii
Chapter 5(cid:4) Differentiation 129
5.1 THEDERIVATIVE 129
5.1.1 The Rules of Differentiation 131
5.1.2 The Chain Rule 133
5.2 THEMEANVALUETHEOREM 135
5.2.1 L’Hôpital’s Rule 139
5.2.2 The Intermediate Value Theorem for Derivatives 143
5.2.3 Inverse Function Theorems 144
5.3 TAYLOR’STHEOREM 147
Chapter 6(cid:4) Riemann Integration 151
6.1 THERIEMANNINTEGRAL 151
6.1.1 Partitions and Darboux Sums 152
6.1.2 Basic Results Regarding Darboux Sums 153
6.1.3 The Definition of the Riemann Integral 156
6.1.4 A Necessary and Sufficient Condition 158
6.2 PROPERTIESOFTHERIEMANNINTEGRAL 162
6.2.1 Linearity Properties 162
6.2.2 Order Properties 165
6.2.3 Integration over Subintervals 166
6.2.4 The Composition Theorem 168
6.3 FAMILIESOFINTEGRABLEFUNCTIONS 170
6.3.1 Continuous Functions 170
6.3.2 Monotone Functions 172
6.3.3 Functions of Bounded Variation 172
6.4 THEFUNDAMENTALTHEOREMOFCALCULUS 177
6.4.1 Evaluating Riemann Integrals 177
6.4.2 Continuous Functions have Antiderivatives 178
6.4.3 Techniques of Antidifferentiation 180
6.4.4 Improper Integrals 184
Chapter 7(cid:4) Infinite Series 189
7.1 CONVERGENCEANDDIVERGENCE 189
7.2 CONVERGENCETESTS 196
viii (cid:4) Contents
7.2.1 Comparison Tests 196
7.2.2 The Integral Test 198
7.2.3 Alternating Series 199
7.2.4 Absolute Convergence 200
7.2.5 The Ratio and Root Tests 201
7.3 REGROUPINGANDREARRANGINGTERMSOFASERIES 205
7.3.1 Regrouping 206
7.3.2 Rearrangements 207
Chapter 8(cid:4) Sequences and Series of Functions 213
8.1 POINTWISEANDUNIFORMCONVERGENCE 213
8.1.1 Sequences of Functions 213
8.1.2 Series of Functions 216
8.2 PRESERVATIONTHEOREMS 218
8.3 POWERSERIES 224
8.4 TAYLORSERIES 230
Appendix A(cid:4) Proof of the Composition Theorem 235
Appendix B(cid:4) Topology on the Real Numbers 239
B.1 OPENANDCLOSEDSETS 239
B.1.1 Continuity Revisited 241
B.1.2 Accumulation Points Revisited 242
B.2 COMPACTSETS 244
B.3 THEHEINE–BORELTHEOREM 247
B.3.1 The Finite Intersection Property 249
B.3.2 The Cantor Set 249
Appendix C(cid:4) Review of Proof and Logic 253
Bibliography 261
List of Symbols 263
Index 265
Preface
Realanalysisistheimportantbranchofmathematicsthatinvestigatestheproperties
oftherealnumbersandestablishesthetheorybehindcalculus,differentialequations,
probability, and other related subjects. The main concepts studied in real analysis
are sets of real numbers, functions, limits, sequences, continuity, differentiation, in-
tegration, and sequences of functions. The study of these topics allows one to gain a
much deeper understanding of the behavior and properties of real-valued functions,
sequences, and sets of real numbers. Such depth allows one to really understand why
the theorems of the calculus are true.
One of the main goals of a course in real analysis is to cover the proofs that
were omitted in calculus. Calculus books usually prove a few of the easiest theorems.
However, if you look carefully, you will see that most calculus textbooks do not
prove, in the main body of the text, many of the most important theorems in the
calculus. The Intermediate Value Theorem, the Extreme Value Theorem, and the
integrabilityofcontinuousfunctionsonaclosedboundedintervalareresultsallcrucial
to calculus and to higher analysis. Moreover, these theorems can be stated in terms
that a calculus student can understand, but an actual proof cannot be achieved
without addressing a fundamental question: What are the important properties of
the set of real numbers? This text provides an answer to this question.
Typically, undergraduates view real analysis as one of the most difficult courses
that a mathematics major is required to take. The main reason for this perception is
twofold: One must comprehend new abstract concepts and learn to deal with these
concepts on a level of rigor and proof not previously encountered. In particular, for
many of these students, the mental gymnastics required to prove theorems about
limits is initially formidable. They struggle to find the appropriate values (e.g., δ or
N) that are necessary to compose a logically correct proof concerning the limit of a
function or a sequence. This text offers a resolution to this difficulty. The book not
only presents the central theorems of real analysis, but also shows the reader how to
compose and produce the proofs of these theorems. This approach should be to the
benefit of any undergraduate reader.
In real analysis, a facility for working with the supremum of a bounded set, the
limit of a sequence, and the ε-δ definitions of continuity are essential for a student to
be successful. This book is designed to show students how to overcome their initial
hurdlesandgainsuchafacility.Ipresentproofstrategiesthatexplicitlyshowstudents
how to deal with the fundamental definitions that one encounters in real analysis;
eachofwhichisfollowedbynumerousexamplesofproofsthatusethesestrategies.In
Chapters2–4,manyoftheintroductoryproofsareprecededbya“proofanalysis”that
carefully explains how to apply these strategies. The proof analysis is then followed
ix