Let $c_0$ be the sequences with $\lim_{n\rightarrow \infty} = 0$. Show that the closed unit ball $\{x\in c_0, \|x\| \leq 1\}$ is not compact in $\ell^\infty$.
I know a lemma that says that the infinite closed unit ball are not compact in infinite-dimensional normed spaces.
This seems strange to me. Does not all the subsequences of sequences in $c_0$ converges?