I am trying to solve the following question:
Let $M_1, M_2 \subset M$ be finite length submodules of a module $M$ (where $M$ need not be of finite length). Show that $ \ M_1 + M_2 = \{x_1 + x_2 \in M \, | \, x_i \in M_i\}$ and $M_1 \cap M_2$ have finite length and that $l(M_1) +l(M_2) = l(M_1 \cap M_2) + l(M_1 + M_2).$
I am really not sure how to prove this and was hoping someone could help me?