Suppose $V$ is a vector space of dimension $n$ and $S$ is a subset of $V$ such that $\operatorname{span}(S)=V$. Prove there exists a basis for $V$ in $S$ without assuming $S$ is a finite set.
I'm not sure what direction to take when $S$ is infinite. I know a bunch of facts. I ultimately want to show that it's possible to pick a finite subset of $S$ that generates $V$ and is linearly independent.