### Rohan Ghuge's Math 508 Notes Fall 2017

- Lecture 3, September 5, 2017 -- The natural numbers and countability

- Lecture 4, September 7, 2017 -- Axioms for the real numbers, the least upper bound property

- Lecture 5, September 12,
2017 --
**The rationals in the reals**
- Lecture
6,
** **September 14, 2017** -- nth roots**

- Lecture
7,
** **September 19, 2017** -- irrational numbers**

** **- Lecture
8,
** **September 21, 2017** -- limits and subsequences**
- Lecture 9, September 26, 2017 -- limsups, liminfs and the Bolzano-Weierstrass theorem

- Lecture 10, September 28,
2017 --
** Cauchy sequences and the construction of the real numbers (see
also these additional notes)**.

- Lecture 11, October 3,
2017 --
** Summing infinite series.**

- Lecture 12, October 10,
2017 --
**Conditional convergence, partial sum formula.**

- Lecture 13, October 12,
2017 --
**Open and closed sets.**
- Lecture 14, October 17, 2017 --Limit points and the Cantor set

- Lecture 15, October 24, 2017 --Continuous functions

- Lecture 16, October 26, 2017
** --The IVT and comtinuous functions on compact sets (thanks to Luv Iyer)**
- Lecture 17, October 31, 2017
** --Monotone functions. The derivative.**
- Lecture 18, November 2, 2017
** --The Mean Value Theorem and the Inverse Function Theorem**
- Lecture 19, November 7, 2017 -- Inverse FunctionTheorem and Higher Order Derivatives

- Lecture 20, November 9, 2017
** -- Taylor's Theorem and Remainder Formulae**
- Lecture 21, November 14, 2017
** -- Integration of continuous functions and the F.T.C. (thanks to Neda Saleem)**

- Lecture 22, November 16, 2017 -- Introduction to the Riemann Integration
- Lecture 25, November 30, 2017 -- Convergence of Sequences of Functions
- Lecture 26, December 5, 2017 -- Convolution, Approximate Identities

Return to main course web-page