Math 131 - Real Analysis I

Harvey Mudd College - Math 131: Real Analysis I (Spring 2010). Taught by Professor Francis Su, 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.
(from math.hmc.edu)

Constructing the rational numbers


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

References
Math 131: Real Analysis I
Instructor: Professor Francis Su. This course is a rigorous analysis of the real numbers, as well as an introduction to writing and communicating mathematics well.