Preface v
1 The Real Numbers 1
1.1 Discussion: The Irrationality of √2 . . . . . . . . . . . . . . . . . 1
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 13
1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 18
1.5 Cantor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Sequences and Series 35
2.1 Discussion: Rearrangements of Infinite Series . . . . . . . . . . . 35
2.2 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 The Algebraic and Order Limit Theorems . . . . . . . . . . . . . 44
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series . . 50
2.5 Subsequences and the Bolzano–Weierstrass Theorem . . . . . . . 55
2.6 The Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7 Properties of Infinite Series . . . . . . . . . . . . . . . . . . . . . 62
2.8 Double Summations and Products of Infinite Series . . . . . . . . 69
2.9 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Basic Topology of R 75
3.1 Discussion: The Cantor Set . . . . . . . . . . . . . . . . . . . . . 75
3.2 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Perfect Sets and Connected Sets . . . . . . . . . . . . . . . . . . 89
3.5 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Functional Limits and Continuity 99
4.1 Discussion: Examples of Dirichlet and Thomae . . . . . . . . . . 99
4.2 Functional Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Combinations of Continuous Functions . . . . . . . . . . . . . . . 109
4.4 Continuous Functions on Compact Sets . . . . . . . . . . . . . . 114
4.5 The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . 120
4.6 Sets of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous? . . . . . . . . . . . . . . 129
5.2 Derivatives and the Intermediate Value Property . . . . . . . . . 131
5.3 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . 137
5.4 A Continuous Nowhere-Differentiable Function . . . . . . . . . . 144
5.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Sequences and Series of Functions 151
6.1 Discussion: Branching Processes . . . . . . . . . . . . . . . . . . 151
6.2 Uniform Convergence of a Sequence of Functions . . . . . . . . . 154
6.3 Uniform Convergence and Differentiation . . . . . . . . . . . . . 164
6.4 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.5 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.6 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration be Defined? . . . . . . . . . 183
7.2 The Definition of the Riemann Integral . . . . . . . . . . . . . . . 186
7.3 Integrating Functions with Discontinuities . . . . . . . . . . . . . 191
7.4 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . 195
7.5 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 199
7.6 Lebesgue’s Criterion for Riemann Integrability . . . . . . . . . . 203
7.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral . . . . . . . . . . . . . . . . . 213
8.2 Metric Spaces and the Baire Category Theorem . . . . . . . . . . 222
8.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.4 A Construction of R From Q . . . . . . . . . . . . . . . . . . . . 243
Bibliography 251
Index 253