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Supplementary Book
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1. The Elements of Real Analysis, R. G. Bartle, John Wiley and Sons, 1967
2. Real analysis with Real Applications, K. R. Davidson, A. P. Donsig, Prentice Hall, 2002
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Content
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Sets bounded from below and above in real numbers, the least upper bound property of real numbers and its consequences, Archimedean property of real numbers, density of rational and irrational numbers in real numbers, nested closed interval property in real numbers, definition of accumulation points and their propertes, Bolzano-Weierstrass Theorem, real valued sequences and their convergence, Cauchy sequences, Completeness of set of real numbers
Limit superior and limit inferior in set of real numbers, sequences and series of real valued functions and their pointwise limit and uniform convergence, Dini Theorem, uniform convergence of sequences of functions and its integrals and derivatives, Weierstrass M-test for uniform convergence of series, Power series and their properties, definition of metric space and examples of metric spaces, neighborhoods in metric spaces, open and closed sets in metric spaces, sequences in metric spaces, Cauchy sequence in metric spaces, complete metric spaces and examples of complete metric spaces, nowhere dense sets, first and second countable metric spaces, Baire Category Theorem, completion of a metric space, open cover for a metric space, precompact metric space, compact metric spaces and their properties, continuous functions between metric spaces, connected metric spaces, Intermediate Value Theorem
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