Closed Unit ball in $\ell^2$ is not compact?

Denote by $e_n \in \ell^2$ the sequence $(0, \ldots, 0, 1, 0, \ldots)$ with the $1$ at position $n$. Then $$\|e_n - e_m\|_2 = \sqrt 2, \qquad n \ne m $$ Hence $(e_n)$ cannot have any subsequence which is Cauchy. (The closed unit ball is moreover bounded by 1 and closed).

In addition to the above, convexity is straightforward. Pick $x,y$ in the closed unit ball and $0 < t <1$. We need to show that $\|tx + (1-t)y\| \le 1$. This follows from the observation:

$\|tx + (1-t)y\| \le \|tx\| + \|(1-t)y\| = t\|x\| + (1-t)\|y\| \le \max(\|x\|,\|y\|) \le 1 $