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18.100A Real Analysis

18.100A Real Analysis (Fall 2020, MIT OCW). Instructor: Dr. Casey Rodriguez. This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs. (from ocw.mit.edu)

Lecture 06 - The Uncountability of the Real Numbers

Instructor: Dr. Casey Rodriguez. We prove the humble, but powerful, triangle inequality, and show that the real numbers are uncountable, and thus the irrational numbers greatly outnumber the rational numbers. We introduce sequences and limits of real numbers.


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Lecture 01 - Sets, Set Operations and Mathematical Induction
Lecture 02 - Cantor's Theory of Cardinality (Size)
Lecture 03 - Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
Lecture 04 - The Characterization of the Real Numbers
Lecture 05 - The Archimedean Property, Density of the Rotations, and Absolute Value
Lecture 06 - The Uncountability of the Real Numbers
Lecture 07 - Convergent Sequences of Real Numbers
Lecture 08 - The Squeeze Theorem and Operations Involving Convergent Sequences
Lecture 09 - Limsup, Liminf, and the Bolzano-Weierstrass Theorem
Lecture 10 - The Completeness of the Real Numbers and Basic Properties of Infinite Series
Lecture 11 - Absolute Convergence and the Comparison Test for Series
Lecture 12 - The Ratio, Root, and Alternating Series Tests
Lecture 13
Lecture 14 - Limits of Functions in terms of Sequences and Continuity
Lecture 15 - The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function
Lecture 16 - The Min/Max Theorem and Bolzano's Intermediate Value Theorem
Lecture 17 - Uniform Continuity and the Definition of the Derivative
Lecture 18 - Weierstrass' Example of a Continuous and Nowhere Differentiable Function
Lecture 19 - Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem
Lecture 20 - Taylor's Theorem and the Definition of Riemann Sums
Lecture 21 - The Riemann Integral of a Continuous Function
Lecture 22 - Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
Lecture 23 - Pointwise and Uniform Convergence of Sequences of Functions
Lecture 24 - Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits
Lecture 25 - Power Series and the Weierstrass Approximation Theorem