TITLE: $c_0$ is not compact in $\ell^\infty$
QUESTION [5 upvotes]: 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?

REPLY [8 votes]: Let $e_n$ be $0$ except for a $1$ in the $n$th coordinate. Clearly $e_n \in c_0$. Then $\|e_n\|_\infty = 1$, but $\|e_n -e_m \|_\infty = 1$ whenever $n\neq m$. Hence $e_n$ can have no convergent subsequence, hence the set $\{ x \in c_0 | \|x\|_\infty \leq 1 \}$ cannot be compact.
And yes, all subsequences of $c_0$ must converge, since any subsequence of a convergent sequence must converge to the same limit.

REPLY [3 votes]: Define $e_n$ to be the sequence that is zero everywhere except at $n$ where its one. Then $||e_n||=1$ and if we consider the sequence $\{e_n\}_{n=1}^\infty$ then I claim that it has no convergent subsequence. In particular notice that the sequence converges pointwise to $0$, so any limit would have to be $0$. But we have that $||e_n||=1$ so it can't converge in norm to $0$. Thereby it has no convergent subsequence.
Notice that the question is asking about a convergent subsequence of a sequence of sequences. Not a convergent subsequence of an element of $C_0$.