## A Guide to Real VariablesA Guide to Real Variables is an aid and conceptual support for students taking an undergraduate course on real analysis. It focuses on concepts, results, examples and illustrative figures, rather than the details of proofs, in order to remain a concise guide which students can dip into. The core topics of a first real analysis course are covered, including sequences, series, modes of convergence, the derivative, the integral and metric spaces. The next book in this series, Folland's A Guide to Advanced Real Analysis is designed to naturally follow on from this book, and introduce students to graduate level real analysis. Together these books provide a concise guide to the subject at all levels, ideal for student preparation for exams. |

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### Contents

Sequences | 13 |

Series | 23 |

The Topology of the Real Line | 41 |

Limits and the Continuity of Functions | 55 |

The Derivative | 71 |

The Integral | 85 |

Sequences and Series of Functions | 103 |

Advanced Topics | 115 |

Glossary of Terms from Real Variable Theory | 129 |

141 | |

About the Author | 147 |

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### Common terms and phrases

absolutely convergent accumulation point boundary points bounded variation calculate called Cantor set cardinality Cauchy criterion Cauchy sequence closed interval closed set compact set conclude continuous function continuously differentiable Corollary countable define Definition denote dense derivative differentiable function discontinuity element equibounded Example exists f and g Figure finite subcovering fj(x function with domain functions fj converge infimum interval a,b inverse image isolated point Lemma Let f lim f(x limit function limit supremum lower bound Mathematical Mean Value Theorem metric space monotonically increasing function natural numbers non-empty number system one-to-one open covering open interval open set partial sums partition polynomials Proposition Ratio Test rational numbers real analysis real numbers result Riemann integrable Riemann sum Riemann-Stieltjes integral Root Test Ross Honsberger sequence of functions sequence of partial sequence xj series converges series of functions sinx subset supremum uncountable uniform convergence uniformly continuous unit interval Weierstrass write x-+P