Let $L$, $M$, and $N$ are subspace of some vector space. Show that following equation need not be true.
$\begin{equation} L \cap (M + N) = (L \cap M) + (L \cap N) \end{equation}$.
I am trying to prove it by using 'if belongs to LSH then show that it belongs to RHS' argument. I start with the assumption that $v_1 \in M$ and $v_2 \in N$; therefore $v_1 + v_2 \in M+N$ and $v_1 + v_2 \in L$ i.e. $v_1 + v_2 \in LHS$. Now if $v_1 + v_2$ is to belong to RHS, it must happen that $v_1 (v_2) \in (L \cap M)$ and $v_2 (v_1) \in (L \cap N)$ (right?); therefore $v_1$ and $v_2$ both must belong to $L$. This may not be the case.
Is this proof correct? I am confused because the equation looks true and I can not cook-up an example which shows that this may not be the case.