Table of Contents
Preface ix
1 Introduction 1
1.1 Basic Terminology 1
1.2 Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets 7
1.3 Distance and Convergence 10
1.4 Minicourse in Basic Logic 15
1.5 Limit Points and Closure 21
Review Problems for Chapter 1 23
2 Some Basic Topological Properties of Rp 25
2.1 Unions and Intersections of Open and Closed Sets 25
2.2 Compactness 28
2.3 Some Applications of Compactness 33
2.4 Least Upper Bounds and Completeness 37
Review Problems for Chapter 2 42
3 Upper and Lower Limits of Sequences of Real Numbers 45
3.1 Generalization of the Limit Concept 45
3.2 Some Properties of Upper and Lower Limits 49
3.3 Convergence of Power Series 52
Review Problems for Chapter 3 56
4 Continuous Functions 57
4.1 Continuity: Ideas, Basic Terminology, Properties 57
4.2 Continuity and Compactness 64
4.3 Types of Discontinuities 70
4.4 The Cantor Set 76
Review Problems for Chapter 4 80
5 Differentiation 81
5.1 The Derivative and Its Basic Properties 81
5.2 Additional Properties of the Derivative; Some Applications of the Mean Value Theorem 86
Review Problems for Chapter 5 92
6 Riemann-Stieltjes Integration 93
6.1 Definition of the Integral 93
6.2 Properties of the Integral 98
6.3 Functions of Bounded Variation 106
6.4 Some Useful Integration Theorems 111
Review Problems for Chapter 6 115
7 Uniform Convergence and Applications 117
7.1 Pointwise and Uniform Convergence 117
7.2 Uniform Convergence and Limit Operations 122
7.3 The Weierstrass M-test and Applications 125
7.4 Equicontinuity and the Arzela-Ascoli Theorem 130
7.5 The Weierstrass Approximation Theorem 134
ReviewProblems for Chapter 7 139
8 Further Topological Results 141
8.1 The Extension Problem 141
8.2 Baire Category Theorem 144
8.3 Connectedness 150
8.4 Semicontinuous Functions 152
Review Problems for Chapter 8 158
9 Epilogue 161
9.1 Some Compactness Results 161
9.2 Replacing Cantor's Nested Set Property 164
9.3 The Real Numbers Revisited 165
Solutions to Problems 167
Index 207
