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18.100B Real Analysis

18.100B Real Analysis (Spring 2025, MIT OCW). Instructor: Prof. Tobias Holck Colding. This course gives an introduction to analysis, and the goal is twofold:

1. To learn how to prove mathematical theorems in analysis and how to write proofs.

2. To prove theorems in calculus in a rigorous way.

The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riemann integrals. After that, we will move on to differential equations. (from ocw.mit.edu)

Lecture 10 - Continuous Functions; Exponential Function (cont.)

We conclude our proof of the basic properties of the exponential function. This includes a result about products of two series.

Go to the Course Home or watch other lectures:

Lecture 01 - Introduction to Real Numbers

Lecture 02 - Introduction to Real Numbers (cont.)

Lecture 03 - How to Write a Proof; Archimedean Property

Lecture 04 - Sequences; Convergence

Lecture 05 - Monotone Convergence Theorem

Lecture 06 - Cauchy Convergence Theorem

Lecture 07 - Bolzano-Weierstrass Theorem; Cauchy Sequences; Series

Lecture 08 - Convergence Test for Series; Power Series

Lecture 09 - Limsup and Liminf; Power Series; Continuous Functions; Exponential Function

Lecture 10 - Continuous Functions; Exponential Function (cont.)

Lecture 11 - Extreme and Intermediate Value Theorem; Metric Spaces

Review for 18.100B Real Analysis Midterm

Lecture 12 - Convergence in Metric Spaces; Operations on Sets

Lecture 13 - Open and Closed Sets; Coverings; Compactness

Lecture 14 - Sequential Compactness; Bolzano-Weierstrass Theorem in a Metric Space

Lecture 15 - Derivatives; Laws of Differentiation

Lecture 16 - Rolle's Theorem; Mean Theorem; L'Hopital's Rule; Taylor Expansion

Lecture 17 - Taylor Polynomials; Remainder Term; Riemann Integrals

Lecture 18 - Integrable Functions

Lecture 19 - Fundamental Theorem of Calculus

Lecture 20 - Pointwise Convergence; Uniform Convergence

Lecture 21 - Integrals and Derivatives under Uniform Convergence

Lecture 22 - Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)

Lecture 23 - Existence and Uniqueness for ODEs: Picard-Lendelof Theorem

Review for 18.100B Real Analysis Final Exam