## Elements of Real AnalysisElementary Real Analysis is a core course in nearly all mathematics departments throughout the world. It enables students to develop a deep understanding of the key concepts of calculus from a mature perspective. Elements of Real Analysis is a student-friendly guide to learning all the important ideas of elementary real analysis, based on the author's many years of experience teaching the subject to typical undergraduate mathematics majors. It avoids the compact style of professional mathematics writing, in favor of a style that feels more comfortable to students encountering the subject for the first time. It presents topics in ways that are most easily understood, yet does not sacrifice rigor or coverage. In using this book, students discover that real analysis is completely deducible from the axioms of the real number system. They learn the powerful techniques of limits of sequences as the primary entry to the concepts of analysis, and see the ubiquitous role sequences play in virtually all later topics. They become comfortable with topological ideas, and see how these concepts help unify the subject. Students encounter many interesting examples, including "pathological" ones, that motivate the subject and help fix the concepts. They develop a unified understanding of limits, continuity, differentiability, Riemann integrability, and infinite series of numbers and functions. |

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### Contents

Chapter 1 The Real Number System | 1 |

Chapter 2 Sequences | 49 |

Chapter 3 Topology of the Real Number System | 137 |

Chapter 4 Limits of Functions | 177 |

Chapter 5 Continuous Functions | 225 |

Chapter 6 Differentiable Functions | 297 |

Chapter 7 The Riemann Integral | 357 |

Chapter 8 Infinite Series of Real Numbers | 453 |

Chapter 9 Sequences and Series of Functions | 541 |

Logic and Proofs | 583 |

Sets and Functions | 613 |

Answers and Hints for Selected Exercises | 635 |

709 | |

Glossary of Symbols | 719 |

727 | |

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algebra of limits Archimedean calculus Cantor set Cauchy product Cauchy sequence closed interval closed sets cluster point contains continuous at xq contradiction converges absolutely converges uniformly countable decimal decreasing define Definition denote derivative differentiable discontinuity diverges element Example EXERCISE SET exists F F F f(xn fc=i finite fn(x following theorem function f(x Hence improper integral inequality infinite L'Hopital's rule Lemma lim f(x lim xn Maclaurin series mathematical induction mean value theorem monotone increasing n^oo natural numbers neighborhood nonempty open interval ordered field partial sums partition pointwise power series Prove that lim Prove Theorem rational numbers real analysis Riemann integrable Section sequence xn sequential criterion series converges sinx strictly increasing subset Suppose f Taylor's theorem uniformly continuous upper bound Vn G N Vx G R