Elementary Real Analysis



ELEMENTARY REAL ANALYSIS

Second Edition (2008)

-------------------------- Thomson?Bruckner2 --------------------------

Brian S. Thomson

Judith B. Bruckner Andrew M. Bruckner



Thomson*Bruckner*Bruckner

Elementary Real Analysis, 2nd Edition (2008)



This version of Elementary Real Analysis, Second Edition, is a hypertexted pdf file, suitable for on-screen viewing. For a trade paperback copy of the text, with the same numbering of Theorems and Exercises (but with different page numbering), please visit our web site.

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c This second edition is a corrected version of the text Elementary Real Analysis originally published by Prentice Hall (Pearson) in 2001. The authors retain the copyright and all commercial uses.

Original Citation: Elementary Real Analysis, Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner. Prentice-Hall, 2001, xv 735 pp. [ISBN 0-13-019075-61]

Cover Design and Photography: David Sprecher

Date PDF file compiled: June 1, 2008

Trade Paperback published under ISBN 1-434841-61-8

Thomson*Bruckner*Bruckner

Elementary Real Analysis, 2nd Edition (2008)



CONTENTS

PREFACE

VOLUME ONE

1 PROPERTIES OF THE REAL NUMBERS 1.1 Introduction 1.2 The Real Number System 1.3 Algebraic Structure 1.4 Order Structure 1.5 Bounds 1.6 Sups and Infs 1.7 The Archimedean Property 1.8 Inductive Property of IN 1.9 The Rational Numbers Are Dense 1.10 The Metric Structure of R 1.11 Challenging Problems for Chapter 1

Thomson*Bruckner*Bruckner

Elementary Real Analysis, 2nd Edition (2008)

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1

1 1 2 6 10 11 12 16 18 20 22 25

iii

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Notes

27

2 SEQUENCES

29

2.1 Introduction

29

2.2 Sequences

31

2.2.1 Sequence Examples

33

2.3 Countable Sets

37

2.4 Convergence

41

2.5 Divergence

47

2.6 Boundedness Properties of Limits

49

2.7 Algebra of Limits

52

2.8 Order Properties of Limits

60

2.9 Monotone Convergence Criterion

66

2.10 Examples of Limits

72

2.11 Subsequences

78

2.12 Cauchy Convergence Criterion

84

2.13 Upper and Lower Limits

87

2.14 Challenging Problems for Chapter 2

95

Notes

98

3 INFINITE SUMS

103

3.1 Introduction

103

3.2 Finite Sums

105

3.3 Infinite Unordered sums

112

3.3.1 Cauchy Criterion

114

3.4 Ordered Sums: Series

120

3.4.1 Properties

122

3.4.2 Special Series

123

Thomson*Bruckner*Bruckner

Elementary Real Analysis, 2nd Edition (2008)



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3.5 Criteria for Convergence

132

3.5.1 Boundedness Criterion

132

3.5.2 Cauchy Criterion

133

3.5.3 Absolute Convergence

135

3.6 Tests for Convergence

139

3.6.1 Trivial Test

140

3.6.2 Direct Comparison Tests

140

3.6.3 Limit Comparison Tests

143

3.6.4 Ratio Comparison Test

145

3.6.5 d'Alembert's Ratio Test

146

3.6.6 Cauchy's Root Test

149

3.6.7 Cauchy's Condensation Test

150

3.6.8 Integral Test

152

3.6.9 Kummer's Tests

154

3.6.10 Raabe's Ratio Test

157

3.6.11 Gauss's Ratio Test

158

3.6.12 Alternating Series Test

162

3.6.13 Dirichlet's Test

163

3.6.14 Abel's Test

165

3.7 Rearrangements

172

3.7.1 Unconditional Convergence

174

3.7.2 Conditional Convergence

176

3.7.3 Comparison of

i=1

ai

and

iIN ai

177

3.8 Products of Series

181

3.8.1 Products of Absolutely Convergent Series

184

3.8.2 Products of Nonabsolutely Convergent Series

186

3.9 Summability Methods

189

Thomson*Bruckner*Bruckner

Elementary Real Analysis, 2nd Edition (2008)

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