Learn · Real Analysis

Lecture 13 of 25Video Lectures

Lecture 12: Convergence in Metric Spaces; Operations on Sets

Prof. Tobias Holck Colding · Massachusetts Institute of Technology

At MIT OpenCourseWare Transcript

Credits and licence

From
Real Analysis, Massachusetts Institute of Technology (at MIT OpenCourseWare)
Licence
CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only.
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.
Who can watch
Tweens profiles and older. How courses are placed

All the 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