### Yunfeng Zhang

University of California, Los Angeles
MATH 131A
TEACHING
RESEARCH

### Math 131A - Discussion (Fall 2017)

Location and hours: Geology 4645, T 9-9:50am.

TA: Yunfeng Zhang (zyf@math.ucla.edu)

Office hours: MS 3955, R 9-10am and 3-4pm.

We will discuss the following topics and problems. Section and exercise numbers follow the textbook Elementary Analysis - the Theory of Calculus, 2nd Edition, by K. Ross.

§1. A word on the philosophy of mathematics: David Hilbert's dream, shattered by Kurt Gödel.
1.7, 1.11.

§2. 2.4.

§3. A word on making definitions, building the vocabulary from the axioms.
3.8.

§4. A word on rules of logic and set theory.
A remark on existence statements (that is, statements that involve "there exists/exist"), the most fundamental example of which is the existence of the least upper bound of a bounded-above subset of real numbers, for another example, existence of square root of 2 (or any other positive real number).
4.1 (r): prove that $\cap_{n=1}^\infty(1-\frac{1}{n},1+\frac{1}{n})=\{1\}$, 4.5, 4.9, 4.16.

§8. 8.7 (c), 8.9.
A remark on how to learn to devise and write proofs:
1. Make sure to be able to define every concept clearly from the axioms of real numbers, using the rules of logic and set theory.
2. Collect useful strategies and tricks.
3. Separate the scratch paper from the answer sheet.

§9. 9.12, 9.14, 9.15.

§10. Existence of the limit $\lim (1+\frac{1}{n})^n$, which defines the number $e$.
Definition of $a^x$, $a>1$, $x\in\mathbb{R}$.
Prove that $\lim(1+\frac{1}{n})^p=1$ for all $p>0$.

§11. Let $a$ be a real number and $(a_n)$ be a sequence of real numbers. Assume that for any subsequence $(a_{n_k})$ of $(a_n)$, there exists a subsequence $(a_{n_{k_l}})$ of $(a_{n_k})$ such that $\lim a_{n_{k_l}}=a$. Prove that $\lim a_n=a$.
Let $(r_n)$ be a sequence that enumerates all the rational numbers. How many subsequential limits of $(r_n)$ are there? [Answer]

§14. 14.6, 14.12.

§15. 15.7.

§17. 17.5, 17.6, 17.7, 17.8, 17.11, 17.12, 17.13, 17.14.

§18. 18.10.

§19. Example 3.