= Solution
Let $\|x_n\|\leq C$. For each $i$, <Bessel inequality> gives
$$
|\langle x_n,e_i\rangle|\leq C,
$$
so the $i$th coordinate sequence is bounded. By the <Bolzano-Weierstrass theorem>, choose a subsequence on which the first coordinate converges. From it choose a further subsequence on which the second coordinate converges, and continue.
Take the diagonal subsequence $(x_{\phi(n)})$. For every fixed $i$, its $i$th coordinate converges, and it remains bounded by $C$. The <coordinate criterion for weak convergence in a separable Hilbert space> supplies an $x_\infty\in H$ such that
$$
\boxed{x_{\phi(n)}\rightharpoonup x_\infty}.
$$
This proves the theorem on a <weak subsequence of a bounded Hilbert-space sequence>.
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