For my bachelor thesis I'd like to have a short "elementary" proof that $\operatorname{Ass}(I^n)$ stabilizes for large $n$ if $I$ is a monomial ideal in a polynomial ring $K[x_1, \dots, x_r]$ over some field $.

The papers of Brodmann (Asymptotic stability of $\operatorname{Ass}(M / I^nM)$) and Ratliff (On prime divisors of $I^n$, $n$ large) are frankly beyond me. (I have only very basic commutative algebra knowledge. I know basic properties of primary decomposition and localization, that's kind of it.)

I actually found something that looks promising: McAdam, Eakin, The Asymptotic Ass

However I'd like to avoid the language of graded rings altogether if that's somehow possible.

Looking over the papers of Brodmann and Ratliff I could follow that $I^{n+1}:I=I^n$ for large $n$ implies asymptotic monotony of $\operatorname{Ass}(I^n/I^{n+1})$, which is just what I'm looking for.

The question then becomes, is there an elementary proof that $I^{n+1}:I=I^n$ for large $n$ and $I$ a monomial ideal? (I feel there should be given how easy calculating with monomial ideals is.) I would also be thankful for a conceptual explanation of the proof of this fact in general Noetherian rings as given by Ratliff or Brodmann.

Most standard facts about monomial ideals in this context (formulas for colons, intersections, associated primes of $I$ are monomial primes) are available. Gröbner basis methods are not available.

Thanks in advance for any help you can offer!