Real Analysis

1

Lecture 1: Sets, Set Operations and Mathematical Induction

2

Lecture 2: Cantor's Theory of Cardinality (Size)

3

Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

4

Lecture 4: The Characterization of the Real Numbers

5

Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value

6

Lecture 6: The Uncountabality of the Real Numbers

7

Lecture 7: Convergent Sequences of Real Numbers

8

Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences

9

Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem

10

Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series

11

Lecture 11: Absolute Convergence and the Comparison Test for Series

12

Lecture 12: The Ratio, Root, and Alternating Series Tests

13

Lecture 13: Limits of Functions

14

Lecture 14: Limits of Functions in Terms of Sequences and Continuity

15

Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function

16

Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem

17

Lecture 17: Uniform Continuity and the Definition of the Derivative

18

Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function

19

Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem

20

Lecture 20: Taylor's Theorem and the Definition of Riemann Sums

21

Lecture 21: The Riemann Integral of a Continuous Function

22

Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula

23

Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions

24

Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits

25

Lecture 25: Power Series and the Weierstrass Approximation Theorem