| Preface | |
| How to Write Mathematics for Nonmathematicians and Mathematicians Alike | |
| Introduction | p. 1 |
| Basic Terminology | p. 1 |
| Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets | p. 7 |
| Distance and Convergence | p. 10 |
| Minicourse in Basic Logic | p. 15 |
| Limit Points and Closure | p. 21 |
| Review Problems for Chapter 1 | p. 23 |
| Some Basic Topological Properties of [actual symbols not reproducible] | p. 25 |
| Unions and Intersections of Open and Closed Sets | p. 25 |
| Compactness | p. 28 |
| Some Applications of Compactness | p. 33 |
| Least Upper Bounds and Completeness | p. 37 |
| Review Problems for Chapter 2 | p. 42 |
| Upper and Lower Limits of Sequences of Real Numbers | p. 45 |
| Generalization of the Limit Concept | p. 45 |
| Some Properties of Upper and Lower Limits | p. 49 |
| Convergence of Power Series | p. 52 |
| Review Problems for Chapter 3 | p. 56 |
| Continuous Functions | p. 57 |
| Continuity: Ideas, Basic Terminology, Properties | p. 57 |
| Continuity and Compactness | p. 64 |
| Types of Discontinuities | p. 70 |
| The Cantor Set | p. 76 |
| Review Problems for Chapter 4 | p. 80 |
| Differentiation | p. 81 |
| The Derivative and Its Basic Properties | p. 81 |
| Additional Properties of the Derivative; Some Applications of the Mean Value Theorem | p. 86 |
| Review Problems for Chapter 5 | p. 92 |
| Riemann-Stieltjes Integration | p. 93 |
| Definition of the Integral | p. 93 |
| Properties of the Integral | p. 98 |
| Functions of Bounded Variation | p. 106 |
| Some Useful Integration Theorems | p. 111 |
| Review Problems for Chapter 6 | p. 115 |
| Uniform Convergence and Applications | p. 117 |
| Pointwise and Uniform Convergence | p. 117 |
| Uniform Convergence and Limit Operations | p. 122 |
| The Weierstrass M-test and Applications | p. 125 |
| Equicontinuity and the Arzela-Ascoli Theorem | p. 130 |
| The Weierstrass Approximation Theorem | p. 134 |
| Review Problems for Chapter 7 | p. 139 |
| Further Topological Results | p. 141 |
| The Extension Problem | p. 141 |
| Baire Category Theorem | p. 144 |
| Connectedness | p. 150 |
| Semicontinuous Functions | p. 152 |
| Review Problems for Chapter 8 | p. 158 |
| Epilogue | p. 161 |
| Some Compactness Results | p. 161 |
| Replacing Cantor's Nested Set Property | p. 164 |
| The Real Numbers Revisited | p. 165 |
| Solutions to Problems | p. 167 |
| Index | p. 207 |
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