let $(l^{1}(\mathbb Z), || .||_{1})$ be a norm space where $l^{1}(\mathbb Z):=\{x=(x_{n})_{n \in \mathbb Z}\subset\mathbb C : ||x||_{1}<\infty \}$
a.) Show that $\overline{B_{1}(0)}$ is closed and bounded.
b.) Show that $\overline{B_{1}(0)}$ is not compact.
My thoughts: In terms of $a.)$ proving that it is bounded is easiest. Consider $\overline{B_{1}(0)}:=\{x\in l^{1}(\mathbb Z):||x-0||_{1}\leq1\}$, this means $\exists c > 1 $: $\forall x\in \overline{B_{1}(0)}$: $||x||_{1}<1<c$, so it is bounded.
I'm struggling in terms of closedness, how do I prove that $l^{1}(\mathbb Z)/ (\overline{B_{1}(0)})$ is open?
Next, how can I prove that $\overline{B_{1}(0)}$ is not compact? Any hints appreciated.