Here you can find all 26 lectures of my Real Analysis course at Harvey Mudd College. These lectures were taped in Spring 2010 with the help of Ryan Muller and Neal Pisenti.
The entire course is assembled as a playlist on YouTube. And below are links to individual lectures.
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
The text for the course was Principles of Mathematical Analysis by Walter Rudin, but you do not need the text to follow these lectures. Also, I realize the board is hard to read, so I've supplied some linked lecture notes in the tab above (they may not align perfectly with these lectures, since it's from a different semester.)
