Basic Real Analysis
This expanded second edition presents the fundamentals and touchstone results of real analysis in full rigor, but in a style that requires little prior familiarity with proofs or mathematical language.
The text is a comprehensive and largely self-contained introduction to the theory of real-valued functions of a real variable. The chapters on Lebesgue measure and integral have been rewritten entirely and greatly improved. They now contain Lebesgue’s differentiation theorem as well as his versions of the Fundamental Theorem(s) of Calculus.
With expanded chapters, additional problems, and an expansive solutions manual, Basic Real Analysis, Second Edition is ideal for senior undergraduates and first-year graduate students, both as a classroom text and a self-study guide.
Reviews of first edition:
The book is a clear and well-structured introduction to real analysis aimed at senior undergraduate and beginning graduate students. The prerequisites are few, but a certain mathematical sophistication is required. ... The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact, this textbook can serve as a source of examples and exercises in real analysis.
The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest.
[This text] introduces upper-division undergraduate or first-year graduate students to real analysis.... Problems and exercises abound; an appendix constructs the reals as the Cauchy (sequential) completion of the rationals; references are copious and judiciously chosen; and a detailed index brings up the rear.
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2 Sequences and Series of Real Numbers
3 Limits of Functions
4 Topology of R and Continuity
5 Metric Spaces
6 The Derivative
7 The Riemann Integral
8 Sequences and Series of Functions