Real analysis: an introduction [1 ed.] 0201086468, 9780201086461

Table of contents :
Contents

Chapter 0 Notation and Terminology . . . . . . . . . . . 1
0-1 Sets . . . . . . . . . . . . . . . . . 1
0-2 Algebra of sets . . . . . . . . . . . . . . 2
0-3 Ordered pairs and functions . . . . . . . . . . 4
0-4 Indexed families and sequences . . . . . . . . . 8
0-5 Cartesian products . . . . . . . . . . . . . 9
0-6 Relations . . . . . . . . . . . . . . . . 11
0-7 Algebraic concepts . . . . . . . . . . . . . 13
0-8 Pointwise operations on functions. . . . . . . . . 14
0-9 Intervals . . . . . . . . . . . . . . . . 15

Chapter 1 The Real Number System . . . . . . . . . . . 16
1-1 The first twelve axioms . . . . . . . . . . . . 18
1-2 The integers and the rational numbers . . . . . . . 22
1-3 The completeness axiom . . . . . . . . . . . 30
Problems . . . . . . . . . . . . . . . . 35

Chapter 2 Metric Spaces . . . . . . . . . . . . . . 44
2-1 Definitions and examples . . . . . . . . . . . 45
2-2 Spheres and sequences . . . . . . . . . . . . 48
2-3 Open sets, cluster points and closed sets . . . . . . . 53
2-4 Continuous functions . . . . . . . . . . . . 58
2-5 Compactness . . . . . . . . . . . . . . 64
2-6 Completeness . . . . . . . . . . . . . . 69
Problems . . . . . . . . . . . . . . . . 70

Chapter 3 Real Functions . . . . . . . . . . . . . . 84
3-1 Real sequences . . . . . . . . . . . . . . 84
3-2 Continuous real functions on a metric space. . . . . . 94
3-3 Continuous real functions on a compact metric space . . . 98
3-4 Uniform convergence and the space C(X) . . . . . . 99
Problems . . . . . . . . . . . . . . . . 107

Chapter 4 The Differential Calculus . . . . . . . . . . . 117
4-1 Difierentiability . . . . . . . . . . . . . . 117
4-2 Rolle’s theorem and the first mean-value theorem . . . . 122
4-3 Sequences of functions . . . . . . . . . . . . 125
Problems . . . . . . . . . . . . . . . . 129


Chapter 5 The Riemann Integral . . . . . . . . . . . . 139
5-1 Upper and lower integrals: the Riemann integral . . . . 139
5-2 Conditions for integrability . . . . . . . . . . 143
5-3 Classes of integrable functions. . . . . . . . . . 147
5-4 Basic properties of the Riemann integral . . . . . . 149
5-5 The fundamental theorem . . . . . . . . . . . 159
5-6 Integration by substitution . . . . . . . . . . 162
5-7 Integration of sequences . . . . . . . . . . . 164
5-8 Extensions of the Riemann integral . . . . . . . . 169
Problems . . . . . . . . . . . . . . . . 175

Chapter 6 Infinite Series, Power Series and Some of Their Applications . 187
6-1 Real series . . . . . . . . . . . . . . . 187
6-2 Series of functions . . . . . . . . . . . . . 194
6-3 Power series . . . . . . . . . . . . . . . 196
6-4 The basic functions of analysis . . . . . . . . . 202
Problems . . . . . . . . . . . . . . . . 207

Chapter 7 Differential Equations . . . . . . . . . . . . 216
7-1 Introduction . . . . . . . . . . . . . . . 216
7-2 A fixed-point theorem . . . . . . . . . . . . 218
7-3 Picard’s theorem. . . . . . . . . . . . . . 220
7-4 Extensions of Picard’s theorem . . . . . . . . . 226
7-5 Linear equations. . . . . . . . . . . . . . 229
Problems . . . . . . . . . . . . . . . . 232

References . . . . . . . . . . . . . . . 237
Index of Symbols and Notation . . . . . . . . . 239
Subject Index. . . . . . . . . . . . . . . 241

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