Real Analysis I

Basic properties of real numbers, supremum and infimum, completeness axiom, open and closed sets, compact sets, countable sets. Limits and convergence of sequences, subsequences, Bolzano-Weierstrass theorem, Cauchy sequences, infinite series, double summations, products of infinite series. Limits of functions, continuity, uniform continuity, intermediate value theorem, extreme-value theorem. Differentiability, derivatives, intermediate value property, Cauchy mean value theorem, Taylor's theorem, Lagrange's form of the remainder. Sequence and series of functions, uniform convergence and differentiation. Power series, radius of convergence, local uniform convergence of power series.