This week we learned about continuous functions, and compactness. We will have some homework about proving compactness.

1. In class, we proved that $[0,1]$ is sequentially compact, can you prove that $[0,1]^2$ in $\R^2$ is sequentially compact? (In general, if metric space $X$ and $Y$ are sequentially compact, we can show that $X \times Y$ is sequentially compact.)

2. Let $E$ be the set of points $x \in [0,1]$ whose decimal expansion consist of only $4$ and $7$ (e.g. $0.4747744$ is allowed), is $E$ countable? is $E$ compact?

3. Let $A_1, A_2, \cdots$ be subset of a metric space. If $B = \cup_i A_i$, then $\bar B \supset \cup_i \bar A_i$. Is it possible that this inclusion is an strict inclusion?

4. Last time, we showed that any open subset of $\R$ is a countable disjoint union of open intervals. Here is a claim and argument about closed set: {\em every closed subset of $\R$ is a countable union of closed intervals. Because every closed set is the complement of an open set, and adjacent open intervals sandwich a closed interval.} Can you see where the argument is wrong? Can you give an example of a closed set which is not a countable union of closed intervals? (here countable include countably infinite and finite)

5. Next week we are going to discuss open cover compactness implies sequential compactness. You can read in Pugh chapter 2 section 7.

1-3. Ross 13.3, 13.5, 13.7

4. Recall that in class, given $(X, d)$ a metric space, and $S$ a subset of $X$, we defined the closure of $S$ to be $$ \bar S = \{ p \in X \mid \text{there is a subsequence $(p_n)$in $S$ that converge to $p$\} $$

Prove that taking closure again won't make it any bigger, i.e, if $S_1 = \bar S$, and $S_2 = \bar S_1$, then $S_1 = S_2$.

5. Prove that $\bar S$ is the intersection of all closed subsets in $X$ that contains $S$. (you may assume result in 4, namely, $\bar S$ is closed)

0. List $\geq 5$ concrete/detailed questions that you wanted to ask (but were too afraid to ask), about lecture notes, about homework, about textbook statements or proofs. Post them on your course homepage, then post the link to discord. Try help others

Rudin's exercises are hard, You don't have to solve all of them. Try work through the first few exercises in Ross in each section, those are meant to consolidate the understanding of basics.

1. Ross 12.10, 12.12, 14.2, 14.10

2. Rudin's exercises in Ch 3: 6, 7, 9, 11

1. Ross Ex 10.6, 11.2, 11.3, 11.5
2. How would you explain 'what is limsup'? For example, you can say something about: What's the difference between limsup and sup? What is most counter-intuitive about limsup? Can you state some sentences that seems to be correct, but is actually wrong?

Due on gradescope next Friday 6pm. You should also submit on discord around Wednesday for others to comment on it.

0. Discussion problems: 9.9, 9.15, 10.7, 10.8

1. About recursive sequence: Ross Ex 10.9, 10.10, 10.11

2. Squeeze test. Let $a_n, b_n, c_n$ be three sequences, such that $a_n \leq b_n \leq c_n$, and $L = \lim a_n = \lim c_n$. Show that $\lim b_n = L$.