# linear algebra, proof help, proof that two definitions of the span are equal

I am stuck on this proof. I want to prove this equivalent definition of Sp A. Let $V$ be the underlying vector space.

def 1: $Sp A$ is the set of all finite linear combinations of elements of $A$.

def 2: $Sp A$ is the intersection of the set of all linear subspaces of $V$ which contain $A$.

Let $A_1$ denote the subset given by def 1, and $A_2$ the subset given by def 2.

I am able to do $A_1 \subset A_2$ like this:

$x \in A_1$ implies that x is a linear combination of finitely many vectors of A.

If $B \subset$ V is a linear subspace of $V$ which contains $A$, then it must contain all finite linear combinations of vectors from $A$.

Hence x is in all the linear subspaces of V which contain A, and hence in the intersection which is $A_2$.

But $A_2\subset A_1$ is kind of difficult for me. Here is my try:

If $x \in A_2$, then $x$ is in all the linear subspaces of V which contain A.

How do I now show that $x$ must be a finite linear combination of vectors from $A$?

Simply prove that $\operatorname{Sp} A$ (def 1) is a linear subspace. Then if $v$ is in the definition 2 span it will be in the definition 1 span, since amongst all those spaces contains $A$ is the definition 1 span.
Suppose that $S$ is a set, and $W$ is a subspace that contains it. If $x,y$ are in $S$, they are in $W$; and since $W$ is a subspace, $\alpha x+\beta y\in W$ for any $\alpha,\beta$. Doing this with as many vectors of $S$ as you like, you get that any arbitrary linear combination $\sum \alpha_iv_i$ is in $W$, for any subspace $W$ containing $S$. Hence ${\rm span}(S)\subseteq \bigcap W$. For th converse, simply note ${\rm span}(S)$ is also a linear subspace contianing $S$, so in particular $\bigcap W\subseteq {\rm span}(S)$. This gives equality.