In the construction of a hull in Schlessinger's paper, one small lemma used is not clear in my opinion. That should be stated as follows:
Let $(R,m)$ be a Noetherian complete local ring, $I_1\supset I_2\supset\cdots$ be a descending chain of proper ideals such that $I_n$ contains $m^n$. Let $I$ be the intersection of all $I_n$. Then for every $k\in \mathbb{N}$, there is a $n(k)$ such that $m^k+I$ contains $I_{n(k)}$.
What I did was consider the inverse system $\{(m^k+I_n)/(m^k+I)\}_{n,k}$ and compute the inverse limit in two different ways. $R/m^k$ is Artinian is used and the condition that $I_n$ contains $m^n$ can be dropped. But I feel like there should be more direct ways to prove this. Are there any easier or instant proof for this?