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 $A$?