Corollary

— Let $u \in A[[I]]$; then for $\lim_{n \to \infty} u^n = 0$ to hold it is necessary and sufficient that the constant term of $u$ should be nilpotent.

Let $a_0$ be the constant term of $u$. The constant term of $u^n$ is $\alpha_0^n$, hence the stated condition is necessary; it is sufficient by Prop. 3.