Metric space compactness

Question

Consider the space $C([0,1])$ equipped with the uniform norm. Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.

I'm confused about $\overline{\{g_n\}}$ and how to prove its compactness

Thanks!

Answer 1

To prove that for some given $g_n$, $\overline{\{g_n\}}$ is compact, we can show that the $g_n$ are uniformly bounded and equicontinuous. Arzela-Ascoli then implies that $\{g_n\}$ is precompact in $C([0,1])$, which gives what you want (i.e., compactness of the closure). So, the work will be in finding a sequence of such $g_n$ which do not converge.

Edit: I suppose I used "work" rather loosely above. Combine the given argument with the choice of $g_n$'s suggested and that should do it.

Answer 2

$g_n\equiv 1$ for $n$ even and $g_n\equiv 0$ for $n$ odd.