See P. N. Ruane's review of the third edition, in which he says "I am very impressed with this book by Bartle and Sherbert". The new edition seems to have preserved the book's many virtues.
One of the changes with this edition is that Robert Bartle has passed away; the new edition is dedicated to his memory. Bartle was also the author of a more advanced text, Elements of Real Analysis, which is in the MAA's list of recommendations for undergraduate mathematics libraries. Introduction to Real Analysis is intended as an undergraduate textbook offering a first exposure to (single-variable) real analysis. It even has an appendix on "proofs and logic", though the author argues that "it is a more useful experience to learn how to construct proofs by first watching and then doing than by reading about techniques of proof." (I agree!)
In the preface, the author tells us that "this edition maintains the same spirit and user-friendly approach as earlier editions. Every section has been examined. Some sections have been revised, new examples and exercises have been added, and a new section on the Darboux approach to the integral has been added to Chapter 7." (The "Darboux approach" uses upper and lower sums instead of Riemann sums, but the resulting integral is equivalent to the Riemann integral.)
Two interesting features of the earlier editions have been retained. First, notions of topology (metric spaces, compactness, etc.) are postponed until the very last chapter. Second, the authors include an introduction to the generalized Riemann integral (aka the Henstock-Kurzweil or gauge integral).
This is a solid introductory textbook that takes great pains to help students achieve the necessary ability and handling formal arguments. Instructors who are content in staying in a one-variable setting will want to consider it for course adoption.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME and the editor of MAA Reviews.
1.1 Sets and Functions.
1.2 Mathematical Induction.
1.3 Finite and Infinite Sets.
CHAPTER 2 THE REAL NUMBERS.
2.1 The Algebraic and Order Properties of R.
2.2 Absolute Value and the Real Line.
2.3 The Completeness Property of R.
2.4 Applications of the Supremum Property.
CHAPTER 3 SEQUENCES AND SERIES.
3.1 Sequences and Their Limits.
3.2 Limit Theorems.
3.3 Monotone Sequences.
3.4 Subsequences and the Bolzano-Weierstrass Theorem.
3.5 The Cauchy Criterion.
3.6 Properly Divergent Sequences.
3.7 Introduction to Infinite Series.
CHAPTER 4 LIMITS.
4.1 Limits of Functions.
4.2 Limit Theorems.
4.3 Some Extensions of the Limit Concept.
CHAPTER 5 CONTINUOUS FUNCTIONS.
5.1 Continuous Functions.
5.2 Combinations of Continuous Functions.
5.3 Continuous Functions on Intervals.
5.4 Uniform Continuity.
5.5 Continuity and Gauges.
5.6 Monotone and Inverse Functions.
CHAPTER 6 DIFFERENTIATION.
6.1 The Derivative.
6.2 The Mean Value Theorem.
6.3 L’Hospital’s Rules.
6.4 Taylor’s Theorem.
CHAPTER 7 THE RIEMANN INTEGRAL.
7.1 Riemann Integral.
7.2 Riemann Integrable Functions.
7.3 The Fundamental Theorem.
7.4 The Darboux Integral.
7.5 Approximate Integration.
CHAPTER 8 SEQUENCES OF FUNCTIONS.
8.1 Pointwise and Uniform Convergence.
8.2 Interchange of Limits.
8.3 The Exponential and Logarithmic Functions.
8.4 The Trigonometric Functions.
CHAPTER 9 INFINITE SERIES.
9.1 Absolute Convergence.
9.2 Tests for Absolute Convergence.
9.3 Tests for Nonabsolute Convergence.
9.4 Series of Functions.
CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL.
10.1 Definition and Main Properties.
10.2 Improper and Lebesgue Integrals.
10.3 Infinite Intervals.
10.4 Convergence Theorems.
CHAPTER 11 A GLIMPSE INTO TOPOLOGY.
11.1 Open and Closed Sets in R.
11.2 Compact Sets.
11.3 Continuous Functions.
11.4 Metric Spaces.
APPENDIX A LOGIC AND PROOFS.
APPENDIX B FINITE AND COUNTABLE SETS.
APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA.
APPENDIX D APPROXIMATE INTEGRATION.
APPENDIX E TWO EXAMPLES.
HINTS FOR SELECTED EXERCISES.