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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 09 - Limsup, Liminf, and the Bolzano-Weierstrass Theorem

Instructor: Dr. Casey Rodriguez. Does a bounded sequence have a convergent subsequence? We introduce limit inferiors and limit superiors to prove this is the case (known as the Bolzano-Weierstrass theorem).


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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