Introduction to Real Analysis, 3rd Edition,9780471321484

Introduction to Real Analysis, 3rd Edition

by ;
Edition: 3rd
ISBN13: 9780471321484
ISBN10: 0471321486
Format: Hardcover
Pub. Date: 1999-10-01
Publisher(s): Wiley
List Price: $194.66

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Summary

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

CHAPTER 1 PRELIMINARIES
1(21)
1.1 Sets and Functions
1(11)
1.2 Mathematical Induction
12(4)
1.3 Finite and Infinite Sets
16(6)
CHAPTER 2 THE REAL NUMBERS
22(30)
2.1 The Algebraic and Order Properties of R
22(9)
2.2 Absolute Value and Real Line
31(3)
2.3 The Completeness Property of R
34(4)
2.4 Applications of the Supremum Property
38(6)
2.5 Intervals
44(8)
CHAPTER 3 SEQUENCES AND SERIES
52(44)
3.1 Sequences and Their Limits
53(7)
3.2 Limit Theorems
60(8)
3.3 Monotone Sequences
68(7)
3.4 Subsequences and the Bolzano-Weierstrass Theorem
75(5)
3.5 The Cauchy Criterion
80(6)
3.6 Properly Divergent Sequences
86(3)
3.7 Introduction to Series
89(7)
CHAPTER 4 LIMITS
96(23)
4.1 Limits of Functions
97(8)
4.2 Limit Theorems
105(6)
4.3 Some Extensions of the Limit Concept
111(8)
CHAPTER 5 CONTINUOUS FUNCTIONS
119(38)
5.1 Continuous Functions
120(5)
5.2 Combinations of Continuous Functions
125(4)
5.3 Continuous Functions on Intervals
129(7)
5.4 Uniform Continuity
136(9)
5.5 Continuity and Gauges
145(4)
5.6 Monotone and Inverse Functions
149(8)
CHAPTER 6 DIFFERENTIATION
157(36)
6.1 The Derivative
158(10)
6.2 The Mean Value Theorem
168(8)
6.3 L'Hospital Rules
176(7)
6.4 Taylor's Theorem
183(10)
CHAPTER 7 THE RIEMANN INTEGRAL
193(34)
7.1 The Riemann Integral
194(8)
7.2 Riemann Integrable Functions
202(8)
7.3 The Fundamental Theorem
210(9)
7.4 Approximate Integration
219(8)
CHAPTER 8 SEQUENCES OF FUNCTIONS
227(26)
8.1 Pointwise and Uniform Convergence
227(6)
8.2 Interchange of Limits
233(6)
8.3 The Exponential and Logarithmic Functions
239(7)
8.4 The Trigonometric Functions
246(7)
CHAPTER 9 INFINITE SERIES
253(21)
9.1 Absolute Convergence
253(4)
9.2 Tests for Absolute Convergence
257(6)
9.3 Tests for Nonabsolute Convergence
263(3)
9.4 Series of Functions
266(8)
CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL
274(38)
10.1 Definition and Main Properties
275(12)
10.2 Improper and Lebesgue Integrals
287(7)
10.3 Infinite Intervals
294(7)
10.4 Convergence Theorems
301(11)
CHAPTER 11 A GLIMPSE INTO TOPOLOGY
312(22)
11.1 Open and Closed Sets in R
312(7)
11.2 Compact Sets
319(4)
11.3 Continuous Functions
323(4)
11.4 Metric Spaces
327(7)
APPENDIX A LOGIC AND PROOFS 334(9)
APPENDIX B FINITE AND COUNTABLE SETS 343(4)
APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 347(4)
APPENDIX D APPROXIMATE INTEGRATION 351(3)
APPENDIX E TWO EXAMPLES 354(3)
REFERENCES 357(1)
PHOTO CREDITS 358(1)
HINTS FOR SELECTED EXERCISES 359(22)
INDEX 381

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