Real Analysis (2020)

26 videos

A first course in Real Analysis, equivalent to an Analysis 1 course — the rigorous rebuilding of single-variable calculus from the axioms up. We open with preliminaries: logic, sets, functions, and then the field and positivity axioms for the real numbers, along with the distribution of the integers and rationals and the standard inequalities and identities. From there the course follows the usual arc. Convergent sequences: the definition of convergence, sequences and sets, the Monotone Convergence Theorem, and the Sequential Compactness Theorem. Continuous functions: continuity, the Extreme Value and Intermediate Value Theorems, uniform continuity, the epsilon-delta criterion, images and inverses, monotone functions, and limits. Differentiation: the algebra of derivatives, differentiating inverses and compositions, the Mean Value Theorem and Cauchy's, and Leibniz notation. Integration: upper and lower integrals, the Archimedes-Riemann Theorem, additivity, monotonicity and linearity, continuity and integrability, and both Fundamental Theorems of Calculus. Proofs throughout — this is the course where the theorems from calculus finally get argued for.