Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.

My question is as follows. Does it follow that $M$ is finitely generated as an $A$-module if and only if $M/A_{>0}M$ is finitely generated as an $A$-module?

Is this well-known? Can I find a proof of this anywhere? Or could anybody supply a proof?