The key is the following result:
If $X$ is a normed space, and $\{x_n\}\subset X$ a sequence which converges weakly to $X$, then $\sup_{n\in\Bbb N}\lVert x_n\rVert<\infty$.
It's a consequence of Baire's categories theorem applied to $X^*$, the topological dual of $X$.
We show that an unbounded set cannot be weakly compact. If $S$ is not bounded, let $\{x_n\}\subset S$ such that $\lVert x_n\rVert\geqslant n$. If $\{x_{n_k}\}$ is a subsequence of $\{x_n\}$, then for each $k$, $\lVert x_{n_k}\rVert\geqslant n_k$. In particular, $\{x_{n_k}\}$ is not bounded hence cannot be weakly convergent.