Proposition 2

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

a) If the infinite product of general factor $x_n$ is convergent and if $\prod_{n=0}^\infty x_n$ is invertible, for every $\varepsilon > 0$, there exists $n_0$ such that, for $n_0 \leq m \leq n$, one has $\left\| \prod_{k=m}^n x_k - e \right\| \leq \varepsilon$.

b) Conversely, if the sequence $(x_n)$ satisfies this condition, the infinite product of general factor $x_n$ is convergent; moreover, if each of the $x_n$ is invertible, $\prod_{n=0}^\infty x_n$ is invertible.

We leave to the reader the development of the proof of this proposition, which is copied step by step from that of th. 1 of IX, p. 79 (the finite parts $L$ of $\mathbf{N}$ which occur in the latter having simply to be replaced by intervals).