Separability of Banach spaces

I mulled repeatedly for some weeks over the fact, why a Banach space is separable if any element can be approximated by a sequence of finite dimensional elements. It has been formulated as , where is a continuous linear operator with finite rank, and this conclusion was never obvious to me. For another time, there was a fact which should be clear and not difficult to show.

Finally, I remembered: Every (finite dimensional) vector space has a (finite) basis. I.e., if , then there is a basis which can be extended as soon as for any . Any basis is countable, and a theorem says that a space is separable iff there is a countable set with , where is the closure of the set of all (finite) linear combinations of elements in .