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.)