I've gotta prove that
$$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$
is compact. I'm just kind of sure that i've got to use sequential compactness, but i don't even know where to start.
As pointed out in the comment, the set $T$ is not closed, but $$\left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| \leqslant \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\}$$ is. We can show that it is precompact by fixing $\varepsilon$. We then take $N$ such that $|\mu_i|\lt\varepsilon$ if $i\geqslant N+1$ and we use compactness of the set $\prod_{i=1}^N[-\mu_i,\mu_i]$ in $\mathbb R^N$.