I am trying to prove that if $R$ is a Noetherian ring then any submodule of any finitely-generated $R$-module is also finitely-generated.
What I have tried: I know that any finitely generated $R$-module is isomorphic to $R^n/N$ for some $n$ and $N$. Since submodules of $R^n/N$ correspond to submodules of $R^n$ containing $N$, and $R/M$ is finitely generated if $R$ is finitely generated, it suffices to show that any submodule of $R^n$ is finitely generated. The condition that every submodule be finitely generated is equivalent to every increasing chain of submodules terminating. Now since $R$ is Noetherian every increasing chain of ideals terminates. I need to use this to show that every increasing chain of submodules in $R^n$ terminates. I am not sure how to show this though.
I would appreciate any different solutions too.