## Principles of mathematical analysis |

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### Contents

THE REAL AND COMPLEX NUMBER SYSTEMS | 1 |

ELEMENTS OF SET THEORY | 21 |

NUMERICAL SEQUENCES AND SERIES | 41 |

Copyright | |

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algebra bounded variation called Cauchy sequence Chap choose compact set completes the proof complex numbers continuous function continuous mapping continuous on a,b convergent sequence converges uniformly Corollary countable set Definition denote derivative diverges Example Exercise exists fc-form finite fn(x Fourier series function defined function g given Hence Hint holds implies inequality infinite integer integral interval inverse Lebesgue Lebesgue integral Let f lim inf lim sup limit point mean value theorem measurable functions metric space monotonic functions neighborhood nonnegative notation obtain open set E C open subset partial sums partition polynomials positive integer power series properties prove rational numbers real function Riemann Riemann integral set E C Rn shows subsequential limit Suppose f trigonometric uniform convergence uniformly continuous upper bound variation on a,b vector space vector-valued functions