Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map

Question

Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show $$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$ where $\mathfrak{m}$ runs over all maximal ideals containing $I$. For $x\in M$, being in the kernel of the canonical map to localization at $\mathfrak{m}$ means there is some $s\in R-\mathfrak{m}$ such that $sx=0$. Krull's Intersection Theorem says the LHS can be annihilated by some $f\in 1+I$, so I can prove LHS is contained in RHS. For the other direction, what I know is given $x\in \text{RHS}$, there is an $s_\mathfrak{m}$ in $R-\mathfrak{m}$ such that $s_\mathfrak{m}x=0$ for each $\mathfrak{m}$ containing $I$. So we have $(1-s_\mathfrak{m})x=x$. I think maybe I can find some $f\in I$ such that $fx=x$ and then we are done. But I'm stuck now. Could anyone give me some advice? Any help would be appreciated.