Introduction to Real Analysis, 3rd Edition

✍ Scribed by Robert G. Bartle, Donald R. Sherbert

• Publisher: Wiley
• Year: 1999
• Tongue: English
• Leaves: 402
• Edition: 3
• Category: Library

✦ Synopsis

In recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers 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 two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral.

✦ Table of Contents

Introduction to Real Analysis......Page 2
PREFACE......Page 6
CONTENTS......Page 10
Section 1.1 Sets and Functions......Page 14
Section 1.2 Mathematical Induction......Page 25
Section 1.3 Finite and Infinite Sets......Page 29
Section 2.1 The Algebraic and Order Properties of R......Page 35
Section 2.2 Absolute Value and the Real Line......Page 44
Section 2.3 The Completeness Property of R......Page 47
Section 2.4 Applications of the Supremum Property......Page 51
Section 2.5 Intervals......Page 57
CHAPTER 3: SEQUENCES AND SERIES......Page 65
Section 3.1 Sequences and Their Limits ·......Page 66
Section 3.2 Limit Theorems......Page 73
Section 3.3 Monotone Sequences......Page 81
Section 3.4 ... Subsequences and the Bolzano-Weierstrass Theorem......Page 88
Section 3.5 The Cauchy Criterion......Page 93
Section 3.6 Properly Divergent Sequences......Page 99
Section 3. 7 Introduction to Infinite Series......Page 102
CHAPTER 4: LIMITS......Page 109
Section 4.1 Limits of Functions......Page 110
Section 4.2 Limit Theorems......Page 118
Section 4.3 Some Extensions of the Limit Conceptt......Page 124
CHAPTER 5: CONTINUOUS FUNCTIONS......Page 132
Section 5.1 Continuous Functions......Page 133
Section 5.2 Combinations of Continuous Functions......Page 138
Section 5.3 Continuous Functions on Intervals......Page 142
Section 5.4 Uniform Continuity......Page 149
Section 5.5 Continuity and Gauges......Page 158
Section 5.6 Monotone and Inverse Functions......Page 162
CHAPTER 6: DIFFERENTIATION......Page 170
Section 6.1 The Derivative......Page 171
Section 6.2 The Mean Value Theorem......Page 181
Section 6.3 L'Hospital's Rules......Page 189
Section 6.4 Taylor's Theorem......Page 196
CHAPTER 7: THE RIEMANN INTEGRAL......Page 206
Section 7.1 Riemann Integral......Page 207
Section 7.2 Riemann Integrable Functions......Page 215
Section 7.3 The Fundamental Theorem......Page 223
Section 7.4 Approximate Integration......Page 232
Section 8.1 Pointwise and Uniform Convergence......Page 240
Section 8.2 Interchange of Limits......Page 246
Section 8.3 The Exponential and Logarithmic Functions......Page 252
Section 8.4 The Trigonometric Functions......Page 259
Section 9.1 Absolute Convergence......Page 266
Section 9.2 Tests for Absolute Convergence......Page 270
Section 9.3 Tests for Nonabsolute Convergence......Page 276
Section 9.4 Series of Functions......Page 279
CHAPTER 10: THE GENERALIZED RIEMANN INTEGRAL......Page 287
Section 10.1 Definition and Main Properties......Page 288
Section 10.2 Improper and Lebesgue Integrals......Page 300
Section 10.3 Infinite Intervals......Page 307
Section 10.4 Convergence Theorems......Page 314
Section 11.1 Open and Closed Sets in R......Page 325
Section 11.2 Compact Sets......Page 332
Section 11.3 Continuous Functions......Page 336
Section 11.4 Metric Spaces......Page 340
APPENDIX A: LOGIC AND PROOFS......Page 347
APPENDIX B: FINITE AND COUNTABLE SETS......Page 356
APPENDIXC: THE RIEMANN ANDLEBESGUE CRITERIA......Page 360
APPENDIX D: APPROXIMATE INTEGRATION......Page 364
APPENDIX E: TWO EXAMPLES......Page 367
REFERENCES......Page 370
PHOTO CREDITS......Page 371
HINTS FOR SELECTED EXERCISES......Page 372
INDEX......Page 394