I have already asked this at math.stackexchange, but since no one answered there after my edit, I decided to try here, although it might be a non-research level question.

The following version of Nakayama's lemma is from Matsumura's Commutative Ring Theory:

Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal s.t. $IM=M$. Then there exists an $a\in A$ with $a\equiv 1\pmod{I}$ and such that $aM=0$. In particular, if $I\subseteq\operatorname{rad}(A)$ we have $M=0$.

After the proof of this via a generalized Cayley-Hamilton, he mentions that the result 'can easily be proved [..] by induction on the number of generators of $M$.' I wonder: how? I tried doing it similarly to the inductive proof of the 'in particular' part, but it didn't work out for me (see MSE for more information on what I think I was doing wrong).

Wouldn't I need to be able to find an $N\subseteq M$, $IN=N$, with fewer generators than $M$ in a somewhat obvious way to use the induction hypothesis? How could I do this?

Thanks in advance!