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"Real Analysis: An Introduction" by A. J. White is a well-organized and comprehensive textbook suitable for undergraduate students studying mathematics or related disciplines. White's writing style is clear and concise, making complex mathematical concepts accessible to students at various levels of mathematical maturity. One of the strengths of this textbook is its comprehensive coverage of fundamental topics in real analysis. It includes discussions on sequences, series, limits, continuity, differentiation, integration, and sequences of functions. Each chapter is accompanied by numerous examples and exercises, allowing students to reinforce their understanding of the material. White takes a rigorous approach to real analysis, providing students with a solid foundation for further study in advanced mathematics. The book is an invaluable resource for undergraduate students seeking to develop a strong understanding of the principles of real analysis. While there may not be many reviews available online, "Real Analysis: An Introduction" is highly recommended by professors and educators in the field of mathematics. Its clarity, comprehensiveness, and rigorous approach make it an excellent choice for undergraduate courses in real analysis. 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