Course description:

This course is a rigorous analysis of the real numbers, as well as an introduction to writing and communicating mathematics well. Topics will include: construction of the real numbers, fields, complex numbers, topology of the reals, metric spaces, careful treatment of sequences and series, functions of real numbers, continuity, compactness, connectedness, differentiation, and the mean value theorem, with an introduction to sequences of functions. It is the first course in the analysis sequence, which continues in Real Analysis II. Taught through Rudin's Principles of Mathematical Analysis. So it's awesome.

Course topics:

Lecture 1: Constructing the rational numbers. Lecture 2: Properties of Q. Lecture 3: Construction of R. Lecture 4: The Least Upper Bound Property. Lecture 5: Complex Numbers. Lecture 6: The Principle of Induction. Lecture 7: Countable and Uncountable Sets. Lecture 8: Cantor Diagonalization, Metric Spaces. Lecture 9: Limit Points. Lecture 10: Relationship b/t open and closed sets. Lecture 11: Compact Sets. Lecture 12: Relationship b/t compact, closed sets. Lecture 13: Compactness, Heine-Borel Theorem. Lecture 14: Connected Sets, Cantor Sets. Lecture 15: Convergence of Sequences. Lecture 16: Subsequences, Cauchy Sequences. Lecture 17: Complete Spaces. Lecture 18: Series. Lecture 19: Series Convergence Tests. Lecture 20: Functions - Limits and Continuity. Lecture 21: Continuous Functions. Lecture 22: Uniform Continuity. Lecture 23: Discontinuous Functions. Lecture 24: The Derivative, Mean Value Theorem. Lecture 25: Taylor's Theorem. Lecture 26: Ordinal Numbers, Transfinite Induction.







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