$\def\m{\mathfrak{m}} $I am trying to show the following commutative algebra result:
Proposition. Let $(R,\m)$ be a noetherian local ring, $M$ be a noetherian module and $M_1,M_2\subset M$ be submodules. If for each non-negative integer $k$ we have $M_1\subset M_2+\m^kM$, then $M_1\subset M_2$.
It was stated to me during a several complex variables lecture without proof, labelled as "Krull's lemma." I have no idea how to prove it. I don't know if I should use Nakayama's lemma in some form or if the lecturer meant the Krull's intersection theorem. The thing is that "krull lemma" doesn't give any results on google.
Does anyone have any hints?