# MAA 5228 Modern Analysis 1

The text for the course is Rudin, *Principles of Mathematical Analysis* (third edition). The topics covered fall into five modules, as follows.

- A review of the real number system, including suprema and infima; a review of the complex number system, including the Cauchy-Schwarz inequality.
- The notions of countability and uncountability. Metric spaces; open sets and closed sets. Compactness; the Heine-Borel and Bolzano-Weierstrass theorems. Connectedness.
- Sequences in metric spaces; subsequences and convergence. Cauchy sequences and complete spaces. Real sequences, including limsup and liminf, with examples. Real and complex series: convergence and absolute convergence.
- Continuous functions between metric spaces. Uniform continuity and continuity on compact spaces. Continuity on connected spaces and the Intermediate Value theorem. Discontinuities and monotonic functions.
- Differentiation of real-valued functions. Fundamental properties, including the chain rule. The Mean Value theorem, with applications including the l’Hopital rule. Higher derivatives and the Taylor theorem.

Note: These modules (in order) closely match the first five chapters of Rudin. Students are encouraged to supplement their study of the text by attempting problems from previous First-year examinations (for which, seehttp://gma.math.ufl.edu/); selections from the past five years or so ought to suffice (though one or two past examinations should be left unseen, for use as practice).