Support turns short exact sequences to unions

Statement
Let $$A$$ be a commutative unital ring. Consider the following short exact sequence of $$A$$-modules:

$$0 \to M \to N \to L \to 0$$.

Then the supports of $$M,N,L$$ are related as follows:

$$Supp(N) = Supp(M) \cup Supp(L)$$

Proof one way
We first show that the support of $$N$$ is contained in the union of the supports of $$M$$ and $$L$$.