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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.
Direct all correspondence to thomson@sfu.ca.
For further information on this title and others in the series visit our website. There are pdf files of the
texts freely available for download as well as instructions on how to order trade paperback copies.
www.classicalrealanalysis.com
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
ClassicalRealAnalysis.com
Thomson*Bruckner*Bruckner
Elementary Real Analysis, 2nd Edition (2008)
CONTENTS
PREFACE xvii
VOLUME ONE 1
1 PROPERTIES OF THE REAL NUMBERS 1
1.1 Introduction 1
1.2 The Real Number System 2
1.3 Algebraic Structure 6
1.4 Order Structure 10
1.5 Bounds 11
1.6 Sups and Infs 12
1.7 The Archimedean Property 16
1.8 Inductive Property of IN 18
1.9 The Rational Numbers Are Dense 20
1.10 The Metric Structure of R 22
1.11 Challenging Problems for Chapter 1 25
iii
ClassicalRealAnalysis.com
Thomson*Bruckner*Bruckner
Elementary Real Analysis, 2nd Edition (2008)
iv
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
ClassicalRealAnalysis.com
Thomson*Bruckner*Bruckner
Elementary Real Analysis, 2nd Edition (2008)
v
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
P
∞
i=1
a
i
and
P
i∈IN
a
i
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
ClassicalRealAnalysis.com
Thomson*Bruckner*Bruckner
Elementary Real Analysis, 2nd Edition (2008)
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