A quick search turns up a proof for finite-dimensional vector spaces:
I think the proof for infinite-dimensional spaces is hard, but don't remember it and can't find one yet. [see below]
Aha- Here's a proof for Hermitian operators that are bounded from below (or above). http://www.ece.rutgers.edu/~maparker/.../Ch03S13ThmsHermOps.pdf
Aha- Here's a good reason why I couldn't remember any proof for infinite-dimensional spaces. http://faraday.uwyo.edu/~yurid/QM/Lecture%208.pdf
They say "In a vector space with a finite dimension, it can be proven rigorously that eigenfunctions of a hermitian transformation span a space, so that any vector can be presented as a linear combination of the basis.
In Hilbert space such a proof exists only for several particular cases."
(published on 02/07/12)