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)