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Course18.100B / 18.100225 lectures

Real Analysis

Prof. Tobias Holck Colding · Massachusetts Institute of Technology

Start with lecture 1 At MIT OpenCourseWare

About

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.

Lectures

Video Lectures

  1. Lecture 1: Introduction to Real Numbers Video
  2. Lecture 2: Introduction to Real Numbers (cont.) Video
  3. Lecture 3: How to Write a Proof; Archimedean Property Video
  4. Lecture 4: Sequences; Convergence Video
  5. Lecture 5: Monotone Convergence Theorem Video
  6. Lecture 6: Cauchy Convergence Theorem Video
  7. Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series Video
  8. Lecture 8: Convergence Tests for Series; Power Series Video
  9. Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function Video
  10. Lecture 10: Continuous Functions; Exponential Function (cont.) Video
  11. Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces Video
  12. Review for 18.100B Real Analysis Midterm Video
  13. Lecture 12: Convergence in Metric Spaces; Operations on Sets Video
  14. Lecture 13: Open and Closed Sets; Coverings; Compactness Video
  15. Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space Video
  16. Lecture 15: Derivatives; Laws for Differentiation Video
  17. Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion Video
  18. Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals Video
  19. Lecture 18: Integrable Functions Video
  20. Lecture 19: Fundamental Theorem of Calculus Video
  21. Lecture 20: Pointwise Convergence; Uniform Convergence Video
  22. Lecture 21: Integrals and Derivatives under Uniform Convergence Video
  23. Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs) Video
  24. Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem Video
  25. Review for the 18.100B Real Analysis Final Exam Video

Syllabus and materials

At MIT OpenCourseWare, as the course publishes them.

Credits and licence

Institution
Massachusetts Institute of Technology
Department
Mathematics
Teacher
Prof. Tobias Holck Colding
Term
Spring 2025
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
MIT OpenCourseWare
Licence
CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only. MIT OpenCourseWare's terms
Attribution
Prof. Tobias Holck Colding. Real Analysis. Spring 2025. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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