## Principles of mathematical analysis |

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### Contents

The Real and Complex Number Systems | 1 |

Elements of Set Theory | 18 |

Numerical Sequences and Series | 36 |

Copyright | |

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apply Theorem bounded variation called Cauchy sequence Chap choose compact set completely additive complex numbers consider contains continuous function continuous on a,b convergent sequence converges absolutely converges uniformly Corollary countable set defined on a,b Definition denote derivative differentiable discontinuous diverges elements equations Example Exercise f is continuous finite set fn(x Fourier series functions defined given Heine-Borel theorem Hence implies inequality infinite subset integer interval Let f lim inf lim sup limit point mean value theorem measurable functions metric space monotonic functions monotonically increasing neighborhood nonnegative notation obtain open set partial sums partition polynomials positive integer power series Proof prove rational number real number system real-valued Riemann integrable segment series converges set of points set of real Suppose f theorem shows trigonometric uniform convergence uniformly continuous upper bound vacuous variation on a,b vector write