Let $I$ be an ideal of a commutative noetherian ring $R$.

How to prove that $\bigcup_{n\ge 1 }\operatorname{Ass}_R (R/I^n)$ is finite ?

I am aware of Brodmann's result about Asymptotic stability of $\operatorname{Ass}_R(M/I^nM)$ for finitely generated module $M$ over commutative Noetherian ring $R$ , but I am trying to prove the claim in a direct, elementary way. I am, of course, allowed to use primary decomposition.

I can see that $Ass_R (\bigoplus_{n \ge 1} R/I^n)=\bigcup_{n\ge 1 }\operatorname{Ass}_R (R/I^n)$, however since $\bigoplus_{n \ge 1} R/I^n$ is not finitely generated, I'm not sure if it is helpful here or not ...

Please help


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