"Let $V$ be a vector space. Let $A$ and $B$ be subsets of $V$.
Let $A+B = \{a+b :a\in A, b\in B\}$
Using the fact that given a vector space $V$, with $S \subseteq V$, then $span(S)$ is the intersection of all subspaces $U$ of $V$, such that $S\subseteq U$, prove it is always true that $span(A+B)\subseteq span(A) + span(B)$"
Where do you begin here?
My idea is that, if I can show that:
A) $A + B \subseteq span(A) + span(B)$
B) $span(A) + span(B)$ is a subspace of $V$
then you can use the fact to prove what we want, as it implies $span(S)\subseteq U$. Is this correct?
Yes, your idea is perfect.
A) easily follows from $S\subseteq span(S)$
B) is easy as well: by definition, $span(S)$ is an intersection of subspaces containing $S$ as a subset, hence itself is a subspace, and all left to prove is that sum of subspaces is again a subspace, i.e. closed under linear combinations.