let $A$ be a noetherian local ring, with maximal ideal $\frak m$. I suppose moreover that $A$ is complete for the $\frak m$-adic topology, if that helps. Let $(I_n)_{n \ge 1}$ be a decreasing sequence of ideals of $A$, with $\bigcap I_n=0$.
Do I have : $$\forall k \ge 1, \exists n \ge 1, I_n \subset \mathfrak{m}^k$$ ?? (i.e. : if the $I_n$ 'tend' to $0$, are they in each $\mathfrak{m}^k$ ?) [Edit : I juste noticed that this is nothing but a theorem by Chevalley, lemma 7 in Chevalley, On the theory of local rings, Ann. Math. 44 (4), 1943]