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Math 131: Real Analysis I

Math 131: Real Analysis I consists of 26 lectures taught by Francis Su, a professor of mathematics at Harvey Mudd College. This course is a rigorous analysis of the real numbers. 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. (from math.hmc.edu)

Lecture 01 - Constructing the rational numbers

Lecture 02 - Properties of Q

Lecture 03 - Construction of the Reals

Lecture 04 - The Least Upper Bound Property

Lecture 05 - Complex Numbers

Lecture 06 - The Principle of Induction

Lecture 07 - Countable and Uncountable Sets

Lecture 08 - Cantor Diagonalization and Metric Spaces

Lecture 09 - Limit Points

Lecture 10 - The Relationship between Open and Closed Sets

Lecture 11 - Compact Sets

Lecture 12 - Relationship of Compact Sets to Closed Sets

Lecture 13 - Compactness and the 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, Absolute Convergence

Lecture 20 - Functions - Limits and Continuity

Lecture 21 - Continuous Functions

Lecture 22 - Uniform Continuity

Lecture 23 - Discontinuous Functions

Lecture 24 - The Derivative and the Mean Value Theorem

Lecture 25 - Taylor's Theorem, Sequence of Functions

Lecture 26 - Ordinal Numbers and Transfinite Induction