Show that if S is a subset of a vector space V, and W is a subset of V where $W \supset S$, then $W \supset Span(S)$

Question

QUESTION

Show that if S is a subset of a vector space V, and W is a subset of V where W ⊃ S, then W ⊃ Span(S)

SOLN

Lemma:
So we have S ⊆ V, W ⊆ V where W ⊃ S, then its easy to prove that Span(W) ⊃ Span(S) because span of W is a linear combination, and every linear combination of vectors of W lies in W. Thus, Span(W) ⊂ W.
Similarly, Span(S) ⊂ S

Where do I go from here and eventually prove that W ⊃ Span(S)?

Answer 1

Your claim isn't true. Consider $V=\mathbb{R}^2$ a vector space over the field $\mathbb{R}$, $W=[-1,1]\times[-1,1]\subseteq\mathbb{R}^2$ and $S=\{(1,1)\}$. Clearly $S\subseteq W$ but $\text{span}(S)\not\subseteq W$. Geometrically, $\text{span}(S)$ is a straight line through the origin.

Answer 2

$W\supset spanW \implies W$ is a subspace... So it's not true in general...