Basic Real Analysis

Week 1 : Review of sequences and series of real numbers.
Week 2 : Tests for convergence of Series. Limit superior and limit inferior.
Week 3 : Cauchy sequences and completeness of R.
Week 4 : Basic notions of Metric Spaces with emphasis on Rn. Connectedness, Compactness, and Heine Borel Theorem.
Week 5 : Continuity and Uniform continuity.
Week 6 : Monotone functions and functions of bounded variation.
Week 7 : Derivatives. Mean Value Theorem and applications
Week 8 : Riemann Stieltjes integral. Riemann`s Criterion for integrability. Improper integrals and the Gamma function.
Week 9 : Sequences and series of functions. Uniform convergence.
Week 10 : Functions of several variables: Directional derivative, partial derivative, total derivative,
Week 11 : Mean Value Theorem, Taylor`s Theorem and applications to Maxima/Minima and convexity.
Week 12 : Double and triple integrals. Statement of Fubini's Theorem and change of variable formula (without proofs) with illustrations.