Lemma: Let $A$ be an $R$-Algebra such that $A$ is a finitely generated $R$-Module. Let $M$ be a finitely generated $A$-module. Show that $M$ is a finitely generated $R$-Module.
Proof:
So things I know is that for any $a\in A$: $a=a_1r_1+...+a_nr_n$ with $a_i \in A$ and $r_i \in R$ and $n\in\mathbb{N}$
For $m\in M$, it'd be $m=m_1b_1+...+m_tb_t$ with $m_j\in M$ and $b_j\in A$ and $t\in\mathbb{N}$. How can I get the connection in order to show this lemma.