Prove that $\mathrm{span}(S) = S$ for a subspace $S$. Prove that if $S$ is a subspace of a vector space $V$, then $\mathrm{span}(S) = S$.
What I tried: I considered using the properties of vector spaces or maybe using an example where $S \subseteq \mathbb{R}^2$, but those strategies didn't amount to much. Any thoughts?

Note that the span of a set $S\subset V$ is the intersection of all subspaces of $V$ that contain $S$, i.e. $$\operatorname{Span}(S) = \bigcap_{S\ \subset\ T\ \leqslant V}T. $$
The intersection of subspaces is again a subspace, so $$\bigcap_{S\ \subset\ T\ \leqslant V}T$$ is a subspace containing $S$, which implies that $$\operatorname{Span}(S)\subset \bigcap_{S\ \subset\ T\ \leqslant V}T. $$   But $\operatorname{Span}(S)$ itself is a subspace containing $S$, therefore
$$\operatorname{Span}(S)\supset \bigcap_{S\ \subset\ T\ \leqslant V}T. $$ 
It follows immediately that if $S\leqslant V$, then $\operatorname{Span}(S)=S$.
