Sets, Set Operations and Mathematical Induction
1:14:22Cantor's Theory of Cardinality (Size)
1:25:07Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
1:18:40The Characterization of the Real Numbers
1:22:04The Archimedian Property, Density of the Rationals, and Absolute Value
1:18:12The Uncountabality of the Real Numbers
1:21:41Convergent Sequences of Real Numbers
1:00:38The Squeeze Theorem and Operations Involving Convergent Sequences
1:14:52Limsup, Liminf, and the Bolzano-Weierstrass Theorem
1:13:43The Completeness of the Real Numbers and Basic Properties of Infinite Series
1:15:37Absolute Convergence and the Comparison Test for Series
1:00:03The Ratio, Root, and Alternating Series Tests
1:00:21Limits of Functions
1:12:54Limits of Functions in Terms of Sequences and Continuity
1:01:14The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function
1:01:58The Min/Max Theorem and Bolzano's Intermediate Value Theorem
1:08:24Uniform Continuity and the Definition of the Derivative
1:12:55Weierstrass's Example of a Continuous and Nowhere Differentiable Function
1:15:36Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem
1:14:27Taylor's Theorem and the Definition of Riemann Sums
52:32The Riemann Integral of a Continuous Function
1:06:35Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
1:12:13Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
This lecture extends Cantor's foundational work on cardinality, progressing to ordered sets and the definition of the real numbers. It rigorously demonstrates the incompleteness of the rational numbers via their failure to satisfy the least upper bound property.
MIT 18.100A Real Analysis, Fall 2020
Cantor's Theory of Cardinality (Size)
Next · Lecture 4The Characterization of the Real Numbers