Here is a question I was wondering. It seems it should be true, but I can't figure it out.

Let $\displaystyle V$ be an $\displaystyle \mathbb{R}$-vector space and let $\displaystyle S \subset V$ be a nonempty subset of $\displaystyle V$. Let $\displaystyle W$ be the vector space generated by $\displaystyle S$. Does $\displaystyle S$ contain a basis for $\displaystyle W$?

I am aware that this is trivial if $\displaystyle V$ is finite-dimensional, but what about in the infinite-dimensional case? I have the feeling that it requires the axiom of choice.