## Real Analysis |

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### Contents

CHAPTER 0PRELIMINARIES SECTION PAGE 1 Sets | 1 |

Functions and Relations | 3 |

Natural Numbers and Integers | 4 |

Copyright | |

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absolutely continuous Baire function basic neighborhood belongs Bernstein polynomial bounded variation Cauchy characteristic function closed interval complete contained continuous functions converges countably additive defined definition denote derivative DF(x directed function element equality holds equivalent everywhere exercise exists f,Jf finite follows Fourier function f function of intervals functions on Ri Hausdorff space Hence hermitian induction inequality integral intervals of continuity L-function Lebesgue-Stieltjes integral Lemma Let f lim f(x lim inf f(x lim sup lim sup f(x limit linear functional linear space lower semicontinuous maximality principle measurable functions measurable sets metric space natural number non-negative nonempty norm notation open interval open set ordered field partially ordered set polynomial positive integer positive number proof prove real numbers real-valued functions Riemann-Stieltjes integrable sequence step-function subdirected function subsets of Ri suppose supremum Theorem topological space U-function uniformly continuous union upper bound