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Learn rigorous mathematical analysis through this comprehensive MIT course that introduces fundamental concepts of real analysis while developing proof-writing skills. Master the theoretical foundations of calculus by exploring real numbers, limits, convergence, and series before progressing to metric spaces, differentiation, and Riemann integration. Develop proficiency in constructing mathematical proofs while examining the Archimedean property, monotone and Cauchy convergence theorems, and the Bolzano-Weierstrass theorem. Study continuous functions, the exponential function, and apply the extreme and intermediate value theorems within metric space frameworks. Investigate open and closed sets, compactness, and sequential compactness before advancing to derivatives and fundamental differentiation laws. Explore Rolle's theorem, the mean value theorem, L'Hôpital's rule, and Taylor expansions with remainder terms. Examine Riemann integrals, integrable functions, and the fundamental theorem of calculus while analyzing pointwise and uniform convergence. Conclude by studying the integration and differentiation of power series and applying analysis techniques to ordinary differential equations, including the Picard-Lindelöf existence and uniqueness theorem.
