# Show inclusion of a sum of subspaces in another subspace

Let $V$ be a vector space, and let $V',V'',W$ be subspaces of $V$. I want to show that

$$(V' \cap W) + (V'' \cap W) \subseteq (V' + V'') \cap W$$

So I take an element $a \in (V' \cap W) + (V'' \cap W)$ and show that it is also an element of $(V' + V'') \cap W$.

I know that $(V' \cap W)$ and $(V'' \cap W)$ are also subspaces of $V$, and that the sum of elements in a given subspace is also inside the subspace. I do not really know how to attack this question, since I usually work with regular sets, and not vector spaces, and presume that the solution to this problem relies on the definition of vector spaces.

Actually the problem is not really homework, but merely an exercise for a course, but I figured it would be best to tag it as homework anyways.

Take a vector $a+b\in (V'\cap W)+(V''\cap W)$. So $a\in V'\cap W$ and $b\in V''\cap W$.
Now in particular, $a\in V'$ and $b\in V''$, so $a+b\in\dots$
Also, $a\in W$ and $b\in W$, so $a+b\in W$ because $\dots$