# On finiteness of $\cup_{n\ge 1 } \operatorname{Ass}_R (R/I^n)$

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 ...