# Real Analysis

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2 STUDENTS

Contents:
Constructing the rational numbers,Properties of Q,Construction of R,The Least Upper Bound Property,Complex Numbers,The Principle of Induction,Countable and Uncountable Sets,Cantor Diagonalization, Metric Spaces, Limit Points,Relationship b/t open and closed sets,Compact Sets,Relationship b/t compact, closed sets,Compactness, Heine-Borel Theorem,Connected Sets, Cantor Sets,Convergence of Sequences,Subsequences, Cauchy Sequences. ,Complete Spaces,Series,Series Convergence Tests,Functions – Limits and Continuity,Continuous Functions,Uniform Continuity,Discontinuous Functions. ,The Derivative, Mean Value Theorem,Taylor’s Theorem,Ordinal Numbers, Transfinite Induction.

### Course Curriculum

 Real Analysis, Lecture 1: Constructing the Rational Numbers Details 1:2:6 Real Analysis, Lecture 2: Properties of Q Details 1:7:38 Real Analysis, Lecture 3: Construction of the Reals Details 1:15:29 Real Analysis, Lecture 4: The Least Upper Bound Property Details 1:12:5 Real Analysis, Lecture 5: Complex Numbers Details 1:7:57 Real Analysis, Lecture 6: Principle of Induciton Details 1:11:58 Real Analysis, Lecture 7: Countable and Uncountable Sets Details 1:13:16 Real Analysis, Lecture 8: Cantor Diagonalization and Metric Spaces Details 1:16:28 Real Analysis, Lecture 9: Limit Points Details 1:12:11 Real Analysis, Lecture 10: The Relationship Between Open and Closed Sets Details 1:13:2 Real Analysis, Lecture 11: Compact Sets Details 1:10:54 Real Analysis, Lecture 12: Relationship of compact sets to closed sets Details 1:15:55 Real Analysis, Lecture 13: Compactness and the Heine-Borel Theorem Details 1:9:29 Real Analysis, Lecture 14: Connected Sets, Cantor Sets Details 1:3:52 Real Analysis, Lecture 15: Convergence of Sequences Details 0:59 Real Analysis, Lecture 16: Subsequences, Cauchy Sequences Details 1:8:1 Real Analysis, Lecture 17: Complete Spaces Details 1:5:23 Real Analysis, Lecture 18: Series Details 1:11:6 Real Analysis, Lecture 19: Series Convergence Tests, Absolute Convergence Details 1:2:58 Real Analysis, Lecture 20: Functions – Limits and Continuity Details 1:4:26 Real Analysis, Lecture 21: Continuous Functions Details 48:19 Real Analysis, Lecture 22: Uniform Continuity Details 1:10:23 Real Analysis, Lecture 23: Discontinuous Functions Details 44:20 Real Analysis, Lecture 24: The Derivative and the Mean Value Theorem Details 1:5:22 Real Analysis, Lecture 25: Taylor’s Theorem, Sequence of Functions Details 1:9:43 Real Analysis, Lecture 26: Ordinal Numbers and Transfinite Induction Details 1:51

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