Review of the real number system: Real number as complete ordered field, Archimedean property. Countable and uncountable sets, Sequences of real numbers, Subsequences, Cauchy sequence, Monotone sequences, Limit inferior, Limit superior.
Metric Spaces: Definition and examples, open and closed sets, convergence of sequences in metric spaces, completeness, Baire category theorem, connectedness, compactness, Heine-Borel theorem, continuity and limit of functions, Uniform continuity, Continuity and compactness, Continuity and connectedness.
Review of differentiation of real valued functions, Taylor’s theorem.
Integration: The Riemann integral and its properties, monotone functions, functions of bounded variation, The Riemann-Stieltjes integral and its properties, Reduction to Riemann integral, integrals of continuous and monotone functions, fundamental theorems of calculus, integration by parts, change of variables formula, Improper integrals.
Revision of series of real numbers, Sequences and series of functions, Pointwise and uniform convergence, Weierstrass M-test, uniform convergence and its relation to continuity, differentiation and integration, Weierstrass approximation theorem, equicontinuous family of functions, Arzela - Ascoli Theorem, overview of Fourier series.
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