Constructing the Rational Numbers
1:02:05Properties of Q
1:07:38Construction of the Reals
1:15:29The Least Upper Bound Property
1:12:04Complex Numbers
1:07:57Principle of Induction
1:11:58Countable and Uncountable Sets
1:13:16Cantor Diagonalization and Metric Spaces
1:16:27Limit Points
1:12:11The Relationship Between Open and Closed Sets
1:13:02Compact Sets
1:10:54Relationship of Compact Sets to Closed Sets
1:15:55Compactness and the Heine-Borel Theorem
1:09:29Connected Sets, Cantor Sets
1:03:52Convergence of Sequences
59:00Subsequences, Cauchy Sequences
1:08:01Complete Spaces
1:05:23Series
1:11:06Series Convergence Tests, Absolute Convergence
1:02:58Functions - Limits and Continuity
1:04:26Continuous Functions
48:19Uniform Continuity
1:10:22Compactness and the Heine-Borel Theorem
This lecture rigorously establishes the Heine-Borel Theorem, proving the equivalence of compactness, closedness, and boundedness for subsets of Euclidean space R^n, a cornerstone of real analysis.
Real Analysis, Spring 2010, Harvey Mudd College, Professor Francis Su. Playlist, FAQ, writing handout, notes available at: http://analysisyawp.blogspot.com/
Relationship of Compact Sets to Closed Sets
Next · Lecture 14Connected Sets, Cantor Sets