Catalog

Real Analysis

Mathematics MIT CC BY-NC-SA 4.0 25 lectures

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.

Syllabus

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

Course materials