Lecture 14 of 25Video Lectures
Lecture 13: Open and Closed Sets; Coverings; Compactness
Prof. Tobias Holck Colding · Massachusetts Institute of Technology
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
- 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
- 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 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
- 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
- Review for the 18.100B Real Analysis Final Exam