Prove that span(S) is the intersection of all subspaces of V containing S . If we consider S' is the intersection of all subspaces of V containing S.Then we have to prove that span (S)⊆S' and S'⊆span(S).
For S'⊆span(S), if u belongs to S', it is a linear combination of vectors in span (S) hence, S'⊆span(S) as u belongs to span (S) also..but for the second part I got stuck.
If $x \in \operatorname{span}(S)$ and $W$ is any subspce containing $S$ then $W$ contains $x$ (because $x$ is a linear combinatinn of elements of $S$). Hence $x$ belongs to the intersection of all such $W$, which is $S'$. Thus $\operatorname{span}(S) \subseteq S$. $S' \subseteq \operatorname{span}(S)$ follows simply from the fact that $\operatorname{span}(S)$ is itself one of the subspaces containing $S$.