Analysis 1

Chapters

1. Introduction and Motivation (ommitted).
2. Prerequisites: Real numbers and supremum property.
3. Sequences: convergence of ~, Cauchy ~, limsup, liminf, Bolzano-Weierstrass.
4. Topology of the real line: open, closed sets. Compactness and sequential compactness in R.
5. Real-valued functions: (uniform) continuity, sequences of functions and their convergence. Dirichlet’s theorem.
6. Differentiability: characterization via Newton-approximation. Classical derivation of product and chain rule. Taylor approximation, Lagrange remainder.
7. The Riemann integral: Riemann sums, formal derivation of all properties of the Riemann integral. Fundamental theorem of Calculus. Integrability of uniform limits of function sequences. Lagrange remainder in integral form.
8. Series: definition of convergence. Convergence criteria of Cauchy, d’Alembert, and root-criterium. Differentiability and integrability of ~: Weierstrass’ M-test. Abel’s theorem.