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.

Author Notes

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.

Chapter 1 Preliminaries	p. 1
1.1 Sets and Functions	p. 1
1.2 Mathematical Induction	p. 12
1.3 Finite and Infinite Sets	p. 16
Chapter 2 The Real Numbers	p. 23
2.1 The Algebraic and Order Properties of R	p. 23
2.2 Absolute Value and the Real Line	p. 32
2.3 The Completeness Property of R	p. 36
2.4 Applications of the Supremum Property	p. 40
2.5 Intervals	p. 46
Chapter 3 Sequences and Series	p. 54
3.1 Sequences and Their Limits	p. 55
3.2 Limit Theorems	p. 63
3.3 Monotone Sequences	p. 70
3.4 Subsequences and the Bolzano-Weierstrass Theorem	p. 78
3.5 The Cauchy Criterion	p. 85
3.6 Properly Divergent Sequences	p. 91
3.7 Introduction to Infinite Series	p. 94
Chapter 4 Limits	p. 102
4.1 Limits of Functions	p. 103
4.2 Limit Theorems	p. 111
4.3 Some Extensions of the Limit Concept	p. 116
Chapter 5 Continuous Functions	p. 124
5.1 Continuous Functions	p. 125
5.2 Combinations of Continuous Functions	p. 130
5.3 Continuous Functions on Intervals	p. 134
5.4 Uniform Continuity	p. 141
5.5 Continuity and Gauges	p. 149
5.6 Monotone and Inverse Functions	p. 153
Chapter 6 Differentiation	p. 161
6.1 The Derivative	p. 162
6.2 The Mean Value Theorem	p. 172
6.3 L'Hospital's Rules	p. 180
6.4 Taylor's Theorem	p. 188
Chapter 7 The Riemann Integral	p. 198
7.1 Riemann Integral	p. 199
7.2 Riemann Integrable Functions	p. 208
7.3 The Fundamental Theorem	p. 216
7.4 The Darboux Integral	p. 225
7.5 Approximate Integration	p. 233
Chapter 8 Sequences of Functions	p. 241
8.1 Pointwise and Uniform Convergence	p. 241
8.2 Interchange of Limits	p. 247
8.3 The Exponential and Logarithmic Functions	p. 253
8.4 The Trigonometric Functions	p. 260
Chapter 9 Infinite Series	p. 267
9.1 Absolute Convergence	p. 267
9.2 Tests for Absolute Convergence	p. 270
9.3 Tests for Nonabsolute Convergence	p. 277
9.4 Series of Functions	p. 281
Chapter 10 The Generalized Riemann Integral	p. 288
10.1 Definition and Main Properties	p. 289
10.2 Improper and Lebesgue Integrals	p. 302
10.3 Infinite Intervals	p. 308
10.4 Convergence Theorems	p. 315
Chapter 11 A Glimpse into Topology	p. 326
11.1 Open and Closed Sets in R	p. 326
11.2 Compact Sets	p. 333
11.3 Continuous Functions	p. 337
11.4 Metric Spaces	p. 341
Appendix A Logic and Proofs	p. 348
Appendix B Finite and Countable Sets	p. 357
Appendix C The Riemann and Lebesgue Criteria	p. 360
Appendix D Approximate Integration	p. 364
Appendix E Two Examples	p. 367
References	p. 370
Photo Credits	p. 371
Hints for Selected Exercises	p. 372
Index	p. 395

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents