Contents

Preface ix

Acknowledgments xiii

Chapter 1. The Construction of Real and Complex Numbers 1

1.1. The Least Upper Bound Property and the Real Numbers 2

1.2. Consequences of the Least Upper Bound Property 4

1.3. Rational Approximation 5

1.4. Intervals 8

1.5. The Construction of the Real Numbers 9

1.6. Convergence in R 13

1.7. Automorphisms of Fields 17

1.8. Complex Numbers 19

1.9. Convergence in C 20

1.10. Independent Projects 24

Chapter 2. Metric and Euclidean Spaces 33

2.1. Introduction 34

2.2. Definition and Basic Properties of Metric Spaces 34

2.3. Topology of Metric Spaces 36

2.4. Limits and Continuous Functions 44

2.5. Absolute Continuity and Bounded Variation in R 51

2.6. Compactness, Completeness, and Connectedness 56

2.7. Independent Projects 64

Chapter 3. Complete Metric Spaces 77

3.1. The Contraction Mapping Theorem and Its Applications to

Differential and Integral Equations 78

3.2. The Baire Category Theorem and the Uniform Boundedness

Principle 79

3.3. Stone-Weierstrass Theorem 82

3.4. The p-adic Completion of Q 85

3.5. Independent Projects 93

v