Real 

Analysis

Dr. Casey Rodriguez

1. Sets, Set Operations, & Mathematical Induction

2. Cantor's Theory of Cardinality 

3. Cantor's Remarkable Theorem of Rational's Lack of the Least Upper Bound Property

4. The Characterisation of Real Numbers

5. The Archimedian Property, Density of the Rationals, & Absolute Value

6. The Uncountability of the Real Numbers

7. Convergent Sequences of Real Numbers

8. The Squeeze Theorem, & Operations Involving Convergent Sequences 

9. Limsup, Liminf, & The Bolzano-Weierstrass Theorem

10. The Completeness of the Real Number & Basic Properties of Infinite Series

11. Absolute Convergence & The Comparison Test for Series

12. The Ratio, Root, & Alternating Series Tests

13. Limit of Functions

14. Limits of Functions in Terms of Sequence and Continuity

15. The Continuity of Sine and Cosine, & The Many Discontinuities of Dirichlet's Theorem

16. The Min/Max Theorem & Bolzano's Intermediate Value Theorem

17. Uniform Continuity & The Definition of the Derivative 

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

19. Differential Rules, Rolle's Theorem, & The Mean Value Theorem

20. Taylor's Theorem, & The Definition of Reimann Sums

21. The Riemann Integral of a Continuous Function 

22. The Fundamental Theorem of Calculus, Integration by Parts, & Change of Variable Formula

23. Pointwise and Uniform Convergence of Sequences of Functions 

24. Uniform Convergence, The Weierstrass M-Test, & Interchanging Limits

25. Power Series, & The Weierstrass Approximation Theorem