Corollary 4

Let $A$ be a complete semi-local Noetherian ring, $m$ a defining ideal of $A$, $E$ a finitely generated $A$-module and $(F_n)$ a decreasing sequence of submodules of $E$ such that $\bigcap_n F_n = 0$. Then, for all $p > 0$, there exists $n > 0$ such that $F_n \subset m^p E$.

As $A$ is a Zariski ring, $E$ is Hausdorff and the $F_n$ are closed under the $m$-adic topology. On the other hand, $E$ is complete (Chapter III, § 2, no. 12, Corollary 1 to Proposition 16). Finally, $E/m^p E$ is a finitely generated module over the ring $A/m^p$, which is Artinian (Corollary 2); then $E/m^p E$ is an Artinian $(A/m^p)$-module and hence an Artinian $A$-module. The corollary then follows from Chapter III, § 2, no. 7, Proposition 8.