Equivalence of definitions of linear span

Question

I have two definition of linear span:

Definition 1

The linear span of a nonempty set $S\subset V$ is defined by $$ \text{span}(S) =\left\{\sum_{i=1}^nx^i v_i \mid n\in\mathbb{N},\ x^i\in\mathbb{F},\ v_i\in S\right\}. $$

Definition 2

Given the subset $S$ of a vector space $V$,

$$ \text{span}(S):=\bigcap\{W \mid (W\text{ subspace of }V)\land (S\subset W)\}. $$

How do I prove that the two definitions are equivalent? A sketch or some idea would be, I hope, enough.

The equivalence is mentioned in the question Formal definition of linear span, but no prove is given.

The books I'm using have the following:

1. Greub1975 speaks only of System of generators for the vector space and does not address the definition of linear span
2. Brown1988 use def. 2 and affirms the equivalence in a Th. 1.15(a). The "Proof" is: "it follows directly from the definition of the linear span."
3. Roman2008 uses def. 1 and I think that in Th. 1.7 he makes the connection to the other definition. But he does not use the expression of intersection of subspaces that are supersets.

Answer 1

Let's call them $\;D_1,\,D_2\;$ (Definition 1, definition 2, corr.) . If you already proved that $\;D_1\;$ is a subspace, then, as it clearly contains $\;S\;$ , $\;D_1\;$ is one of the subspaces appearing in the definition of $\;D_2\;$ , and thus $\;D_2\subset D_1\;$ .

OTOH, any subspace $\;W\;$ containing $\;S\;$ must contain any linear combination of its elements, thus $\;D_1\subset W\;$ ... End the argument now.