Catalog

Real Analysis

Mathematics MIT CC BY-NC-SA 4.0 25 lectures

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.

Syllabus

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

Course materials