Introduction to Real Analysis

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.

Table of Contents:
CHAPTER 1 PRELIMINARIES 1 1.1 Sets and Functions 1 1.2 Mathematical Induction 12 1.3 Finite and Infinite Sets 16 CHAPTER 2 THE REAL NUMBERS 23 2.1 The Algebraic and Order Properties of R 23 2.2 Absolute Value and the Real Line 32 2.3 The Completeness Property of R 36 2.4 Applications of the Supremum Property 40 2.5 Intervals 46 CHAPTER 3 SEQUENCES AND SERIES 54 3.1 Sequences and Their Limits 55 3.2 Limit Theorems 63 3.3 Monotone Sequences 70 3.4 Subsequences and the Bolzano-Weierstrass Theorem 78 3.5 The Cauchy Criterion 85 3.6 Properly Divergent Sequences 91 3.7 Introduction to Infinite Series 94 CHAPTER 4 LIMITS 102 4.1 Limits of Functions 103 4.2 Limit Theorems 111 4.3 Some Extensions of the Limit Concept 116 CHAPTER 5 CONTINUOUS FUNCTIONS 124 5.1 Continuous Functions 125 5.2 Combinations of Continuous Functions 130 5.3 Continuous Functions on Intervals 134 5.4 Uniform Continuity 141 5.5 Continuity and Gauges 149 5.6 Monotone and Inverse Functions 153 CHAPTER 6 DIFFERENTIATION 161 6.1 The Derivative 162 6.2 The Mean Value Theorem 172 6.3 L'Hospital's Rules 180 6.4 Taylor's Theorem 188 CHAPTER 7 THE RIEMANN INTEGRAL 198 7.1 Riemann Integral 199 7.2 Riemann Integrable Functions 208 7.3 The Fundamental Theorem 216 7.4 The Darboux Integral 225 7.5 Approximate Integration 233 CHAPTER 8 SEQUENCES OF FUNCTIONS 241 8.1 Pointwise and Uniform Convergence 241 8.2 Interchange of Limits 247 8.3 The Exponential and Logarithmic Functions 253 8.4 The Trigonometric Functions 260 CHAPTER 9 INFINITE SERIES 267 9.1 Absolute Convergence 267 9.2 Tests for Absolute Convergence 270 9.3 Tests for Nonabsolute Convergence 277 9.4 Series of Functions 281 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 288 10.1 Definition and Main Properties 289 10.2 Improper and Lebesgue Integrals 302 10.3 Infinite Intervals 308 10.4 Convergence Theorems 315 CHAPTER 11 A GLIMPSE INTO TOPOLOGY 326 11.1 Open and Closed Sets in R 326 11.2 Compact Sets 333 11.3 Continuous Functions 337 11.4 Metric Spaces 341 APPENDIX A LOGIC AND PROOFS 348 APPENDIX B FINITE AND COUNTABLE SETS 357 APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360 APPENDIX D APPROXIMATE INTEGRATION 364 APPENDIX E TWO EXAMPLES 367 REFERENCES 370 PHOTO CREDITS 371 HINTS FOR SELECTED EXERCISES 372 INDEX 395

About the Author :
Robert Gardner Bartle was an American mathematician specializing in real analysis. He is known for writing various popular textbooks. Donald R. Sherbert is the author of Introduction to Real Analysis, 4th Edition, published by Wiley.