Elements of Real Analysis

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Jones & Bartlett Learning, Jan 28, 2011 - Mathematics - 737 pages
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Elementary Real Analysis is a core course in nearly all mathematics departments throughout the world. It enables students to develop a deep understanding of the key concepts of calculus from a mature perspective. Elements of Real Analysis is a student-friendly guide to learning all the important ideas of elementary real analysis, based on the author's many years of experience teaching the subject to typical undergraduate mathematics majors. It avoids the compact style of professional mathematics writing, in favor of a style that feels more comfortable to students encountering the subject for the first time. It presents topics in ways that are most easily understood, yet does not sacrifice rigor or coverage. In using this book, students discover that real analysis is completely deducible from the axioms of the real number system. They learn the powerful techniques of limits of sequences as the primary entry to the concepts of analysis, and see the ubiquitous role sequences play in virtually all later topics. They become comfortable with topological ideas, and see how these concepts help unify the subject. Students encounter many interesting examples, including "pathological" ones, that motivate the subject and help fix the concepts. They develop a unified understanding of limits, continuity, differentiability, Riemann integrability, and infinite series of numbers and functions.
 

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Contents

Chapter 1 The Real Number System
1
Chapter 2 Sequences
49
Chapter 3 Topology of the Real Number System
137
Chapter 4 Limits of Functions
177
Chapter 5 Continuous Functions
225
Chapter 6 Differentiable Functions
297
Chapter 7 The Riemann Integral
357
Chapter 8 Infinite Series of Real Numbers
453
Chapter 9 Sequences and Series of Functions
541
Logic and Proofs
583
Sets and Functions
613
Answers and Hints for Selected Exercises
635
Bibliography
709
Glossary of Symbols
719
Index
727
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