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Course18.100A / 18.100125 lectures

Real Analysis

Dr. Casey Rodriguez · Massachusetts Institute of Technology

Start with lecture 1 At MIT OpenCourseWare

About

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs.

Lectures

Lecture Videos

  1. Lecture 1: Sets, Set Operations and Mathematical Induction Video
  2. Lecture 2: Cantor's Theory of Cardinality (Size) Video
  3. Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property Video
  4. Lecture 4: The Characterization of the Real Numbers Video
  5. Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value Video
  6. Lecture 6: The Uncountabality of the Real Numbers Video
  7. Lecture 7: Convergent Sequences of Real Numbers Video
  8. Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences Video
  9. Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem Video
  10. Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series Video
  11. Lecture 11: Absolute Convergence and the Comparison Test for Series Video
  12. Lecture 12: The Ratio, Root, and Alternating Series Tests Video
  13. Lecture 13: Limits of Functions Video
  14. Lecture 14: Limits of Functions in Terms of Sequences and Continuity Video
  15. Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function Video
  16. Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem Video
  17. Lecture 17: Uniform Continuity and the Definition of the Derivative Video
  18. Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function Video
  19. Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem Video
  20. Lecture 20: Taylor's Theorem and the Definition of Riemann Sums Video
  21. Lecture 21: The Riemann Integral of a Continuous Function Video
  22. Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula Video
  23. Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions Video
  24. Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits Video
  25. Lecture 25: Power Series and the Weierstrass Approximation Theorem Video

Syllabus and materials

At MIT OpenCourseWare, as the course publishes them.

Credits and licence

Institution
Massachusetts Institute of Technology
Department
Mathematics
Teacher
Dr. Casey Rodriguez
Term
Fall 2020
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
Dr. Casey Rodriguez. Real Analysis. Fall 2020. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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