I have been doing some revision for my upcoming Topology Exam and have come across an exercise which I have no idea how to do, as all the methods I try seem to fail. Any solutions or methods would be helpful thanks!
(i) Define the closure $\overline{A}$ of a subset A of a normed vector space X to be the intersection of all closed sets in X containing A:
$$\overline{A} := \bigcap \{F:A\subseteq F \subseteq X \text{ and F is closed} \}.$$
Prove that $\overline{A}$ consists precisely of those $x\in X$ such that $inf_{a\in A}\left \| x-a \right \|=0$.
Then for the following three parts, let B be the set of all sequences of scalars $(a_n)_{n\in\mathbb{N}}$ such that only finitely many of the entries of $a_n$ are non-zero.
(i) Is B a closed subset of $(l^{\infty},\left \|\cdot \right \|_{\infty})$?
(ii) What is the closure of B in $(l^{\infty},\left \|\cdot \right \|_{\infty})$?
(iii) Is B an open subset of $(l^1,\left \|\cdot \right \|_1)$?
I have tried to use neighbourhoods and open balls in lots of different ways but can't even seem to prove part (i), so as a result have had zero luck with the other parts. I would just like to now this before my exam as a similar question may come up as these exercises have been recommended for us to revise from. Thanks!
Let $C_A=\{x:inf_{a\in A}\|x-a\|\}=0\}$. Let $C$ be a closed subset which contains $A$ and $x\in C_A$, for every integer $n>0$, there exists $a_n\in A$ such that $d(a_n,x)<1/n$, this implies that $x=lim_na_n$, since $a_n\in A\subset C$ and $C$ is closed, we deduce that $x\in C$ and $C_A\subset C$.
Let $x$ in the intersection of the closed subsets which contains $A$, suppose that $X$ is not in $C_A$, there exists $c>0$ such that $B(x,c)\cap A$ is empty. The complementary of $B(x,c)$ is a closed subset which contains $A$ but not $x$, contradiction.