Elements of Real AnalysisElementary 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, without sacrificing 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. |
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 |
| 709 | |
Glossary of Symbols | 719 |
| 727 | |
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Common terms and phrases
algebra analysis Apply bounded calculus called Cauchy Chapter closed cluster point compact complete Consider constant contains contradiction Corollary course cover criterion decreasing define Definition denote derivative differentiable discontinuity diverges element equal equation equivalent everywhere Example EXERCISE SET exists f is continuous f is integrable fact Figure find finite first function f G N 9 given Hence induction inequality infinite interval irrational Lemma limit mathematical mean mean value theorem monotone increasing neighborhood Note open interval ordered field partition polynomial positive power series principle Proof proposition Prove Theorem radius of convergence rational numbers real number Riemann integrable rule sequence Solution statement subsequence subset Suppose f symbols true Vn G N