*The unit ball of an Hilbert space is sequentially compact for the weak topology.*

People often think that separability is needed here (this is equivalent to the existence of a countable Hilbert basis) and that without it, the result does not hold and one must use the characterization of compactness by open covers together with Tykhonov theorem + Axiom of choice to get a somewhat weaker statement, namely the existence of a cluster point.

But in fact, one can work in the closed separable subspace generated by the elements of the sequence and proceed by diagonal extraction. Everything is zero in the orthogonal of that subspace and nothing has to be done there. 

There are a few applications of that result in ergodic theory.