Principles of mathematical analysis 
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Review: Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics)
User Review  Scott Grizzard  GoodreadsTHE introductory text for Analysis...it is..."nasty, brutish, and short." However, as Churchill once said of democracy, it is the "worst except for all the others." Read full review
Contents
THE REAL AND COMPLEX NUMBER SYSTEMS  1 
ELEMENTS OF SET THEORY  21 
NUMERICAL SEQUENCES AND SERIES  41 
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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 finite fn(x Fourier series function defined function g given Hence Hint holds implies inequality infinite integer integral interval inverse Lebesgue Lebesgue integral Lebesgue measure lim inf lim sup limit point mean value theorem measurable functions metric space monotonic functions neighborhood nonnegative notation obtain onetoone 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 Suppose f trigonometric uniform convergence uniformly continuous upper bound variation on a,b vector space vectorvalued functions write