Corollary

Let A be a complete normed algebra. Then for each $z \in A$ such that $||z|| < 1$, $e - z$ is a unit in A.

The series whose general term is $z^n$ is absolutely convergent, since $||z^n|| \leq ||z||^n$ for $n > 0$, and is therefore convergent, since A is complete (no. 6, Proposition 11).