# A module trapped between two free modules is finitely generated

Let $R$ be a Noetherian integral domain, $M$ and $R$-module and $F_1$, $F_2$ free $R$-modules of rank $n$ for which $$F_1 \subseteq M \subseteq F_2$$ Is it true that $M$ is finitely generated? If so, is there a "slick" proof of this (perhaps using the fact that $F_1, F_2$ are projective)?

Yes, because $F_2$ is necessarily Noetherian, and then so are all of its submodules.
You can even drop the domain condition and the other free module $F_1$.