Closed and Bounded but not compact Let $(C[0,1];d_{\infty})$ the metric space with $C[0,1]$ the continuous functions on $[0,1]$ and $d_{\infty}(f,g)=\max_{x \in[0,1]}|f(x)-g(x)|$
Prove that $$S=\{f:|f(x)| \leq 1\}$$ is closed and bounded but no compact.
My attempt: I've already proved that is bounded using a ball with radius $2$. But I can't figure out how can I prove that is closed and is not compact, I tried with the limit points and the complement but I'm lost.

For each $n\in \mathbb{N}$,
choose a function $f_n:[0,1]\rightarrow[0,1]$ so that $f$ vanishes off of the closed inteerval $[1/(n+1),1/n]$, $0\le f \le 1$,  and so that $f = 1$ in the middle of the interval.  For all $m$, $n$, 
$$d_{\infty}(f_m, f_n) = 1$$
if $m\not=n$.  This precludes compactness.
