Alternative definition for span and proving it is equivalent to the most common one This is a question related to something that I asked here about this alternative definition of span. User hardmath has helped me a lot! Therefore, I can't still understand how to prove the equivalence between the definition:

Span:
  Suppose a vector space $(V,+,\cdot)$, and 
$$S = \{u_1,\cdots,u_n\}$$
(and $S$ is a subset of $V$, not a subspace)
$$[S]=:\cap_{w\subset V, w\supseteq S} W$$
In other words, $[S]$ is, by definition, the intersection of all $W$,
  such that $W$ is a subspace of $V$ and $W$ contains $S$.

And the definition:

Span:
  $$S = \{u_1,\cdots,u_n\}$$
  The span of $S$, denote by $[S]$ is, then:
  $$W = \{\alpha_1u_1+\cdots+\alpha_nu_n|\alpha_i\in\mathbb R\}$$

I am not a lazy person and I'm not asking for anyone to simply solve it for me, if it will bother you. Any hint is wellcome. So, how do I go from the first definition, to the second?
Couldn't find it on wikipedia, neither in my book. 

You must take the intersection in the first definition over subspaces, not all subsets; indeed the intersection over all subsets will just be $S$ itself.
Let's look at one direction of containment. For convenience I will write $S_1$ for the first definition and $S_2$ for the second. (This is not standard notation by any means.) I'll show that $S_1 \subset S_2$. This follows because $S_2$ is one such subspace $W$, so it is one of the sets in the intersection defining $S_1$, so $S_1 \subset S_2$.
Now try to prove $S_2 \subset S_1$. 
