Introduction to Real AnalysisThis 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. |
Contents
| 1 | |
CHAPTER 2 THE REAL NUMBERS | 23 |
CHAPTER 3 SEQUENCES AND SERIES | 54 |
CHAPTER 4 LIMITS | 102 |
CHAPTER 5 CONTINUOUS FUNCTIONS | 124 |
CHAPTER 6 DIFFERENTIATION | 161 |
CHAPTER 7 THE RIEMANN INTEGRAL | 198 |
CHAPTER 8 SEQUENCES OF FUNCTIONS | 241 |
LOGIC AND PROOFS | 348 |
FINITE AND COUNTABLE SETS | 357 |
THE RIEMANN AND LEBESGUE CRITERIA | 360 |
APPROXIMATE INTEGRATION | 364 |
TWO EXAMPLES | 367 |
REFERENCES | 370 |
PHOTO CREDITS | 371 |
HINTS FOR SELECTED EXERCISES | 372 |
CHAPTER 9 INFINITE SERIES | 267 |
CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL | 288 |
CHAPTER 11 A GLIMPSE INTO TOPOLOGY | 326 |
INDEX | 395 |
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Common terms and phrases
8-fine partition A I R absolutely convergent apply arbitrary belongs bijection calculation Cauchy sequence cluster point conclude continuous functions Convergence Theorem convergent sequence converges uniformly countable Darboux integrable defined derivative differentiable divergent elements endpoint establish Exercises for Section exists f and g f is continuous find finite number first follows from Theorem function f Fundamental Theorem gauge given Hence If(x implies infinite inverse let f lim f lim(xn limit Mathematical Induction Mean Value Theorem metric space monotone natural number nonempty obtain open interval open set partial sums properties prove rational numbers reader real numbers result Riemann integrable satisfies sequence of real Squeeze Theorem statement step function subintervals subset sufficiently supremum tagged partition Taylor’s Theorem Theorem Let Triangle Inequality uniform convergence uniformly continuous upper bound