Compactness of $C([0,1])$ I have to verify if the $C([0,1])$, space of all continuous functions defined on interval $[0,1]$ with supremum metric is compact.
As I know, we have to check if every sequence of functions $f_{n}(x)$ has subsequence that $f_{n_{k}}(x)$ is convergent.
In this metric of course conervgence implies uniform convergence, so there won't be a problem of showing continuity of the limit function.
But I really don't know if we can make subsequence convergent.

There is a general result:
Theorem The unit ball in a normed space is compact if and only if the space is finite dimensional.
An immediate consequence of this is that the unit ball in $C([0,1]$ is not compact and hence (as a closed subset of a space being non-compact implies the space itself is not compact) the whole of $C([0,1])$ is non-compact.
