## Introduction to Real Analysis, 4th EditionThis 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. |

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#### LibraryThing Review

User Review - dwarfplanet9 - LibraryThingThis book was used in my Real Analysis course. The subject would be hard to learn from this book alone, but lucky for me I had a great teacher at San Jose State University. For those trying to use the ... Read full review

#### LibraryThing Review

User Review - ssd7 - LibraryThingIntroduction to Real Analysis is easily one of my favorite mathematics textbooks. The explanation is excellent and the in-text examples are interesting. Unlike most mathematics text books I've read ... Read full review

### 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