Real Analysis | Mathematics | MIT OpenCourseWare

Course Info

Instructor

Prof. Tobias Holck Colding

Departments

Mathematics

As Taught In

Spring 2025

Level

Undergraduate
Graduate

Topics

Mathematics

Learning Resource Types

Exams

Lecture Notes

Lecture Videos

Problem Sets

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Lecture Videos

Lecture 1: Introduction to Real Numbers

Lecture 2: Introduction to Real Numbers (cont.)

Lecture 3: How to Write a Proof; Archimedean Property

Lecture 4: Sequences; Convergence

Lecture 5: Monotone Convergence Theorem

Lecture 6: Cauchy Convergence Theorem

Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series

Lecture 8: Convergence Tests for Series; Power Series

Lecture 9: 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

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 for Differentiation

Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion

Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals

Lecture 18: Integrable Functions

Review for 18.100B Real Analysis Midterm

Review for the 18.100B Real Analysis Final Exam

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 & Uniqueness for ODEs: Picard–Lindelöf Theorem

Course Info

Instructor

Prof. Tobias Holck Colding

Departments

Mathematics

As Taught In

Spring 2025

Level

Undergraduate
Graduate

Topics

Mathematics

Learning Resource Types

Exams

Lecture Notes

Lecture Videos

Problem Sets

Download Course

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