Proposition 2

Let $(x_n)$ be a sequence of points in a complete normed algebra $A$.

a) If the infinite product whose general factor is $x_n$ is convergent and if $\prod_{n=0}^\infty x_n$ is a unit in $A$, then for each $\varepsilon > 0$ there exists an integer $n_0$ such that

whenever $n_0 \leq m \leq n$.

b) Conversely, if the sequence $(x_n)$ satisfies this condition, the infinite product with general factor $x_n$ is convergent; and if each of the $x_n$ is a unit in $A$, then $\prod_{n=0}^\infty x_n$ is a unit.

The proof of this proposition follows step by step the proof of Theorem 1, and is left to the reader (the finite subsets $L$ of $N$ in the proof of Theorem 1 are to be replaced by intervals).