Textbook

# Introduction to Real Analysis, 4th Edition

## Description

This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations, and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with additional examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: Introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.
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CHAPTER 1 PRELIMINARIES.

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.

2.5 Intervals.

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.

REFERENCES.

PHOTO CREDITS.

HINTS FOR SELECTED EXERCISES.

INDEX.

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## New To This Edition

• Several new examples have been added to this edition to make the text more up-to-date and relevant
• New exercises have been added throughout to give students more material to practice and solidify their understanding of the material
• Coverage of the Darboux integral has been added in Section 7.4
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• Treatment of Basic Theory of Functions: Detailed and rigorous treatment of the basic theory of functions of one real variable.
• Pacing: Accessible and user-friendly because concepts are developed in a reasonably paced manner with many examples to illustrate the theory.
• Examples and Solutions: Every concept is illustrated by examples and special cases. There is also a wide range of exercises, and hints for some of them are provided in the back of the text.
• Approximation Methods and Numerical Calculation: These are emphasized whenever appropriate, making this text suitable for computer science students.
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Instructors Resources
Wiley Instructor Companion Site
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Purchase Options
Wiley E-Text
Introduction to Real Analysis, 4th Edition
ISBN : 978-0-470-91181-5
416 pages