**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**