Exercise 8, Appendix 2

Let $A$ be a complete normed algebra; now prove that if $(x_n)$ is a sequence of points of $A$ such that, for every permutation $\sigma$ of $\mathbf{N}$, the sequence $(x_{\sigma(n)})$ is multipliable and has an invertible product, each of the $x_n$ is invertible (one will first establish the following algebra lemma: in a ring having a unit element, if two elements $x, y$ are such that $xy$ is invertible, and $yx$ is not a divisor of 0, each of the elements $x, y$ is invertible; cf. A, I, p. 152, exerc. 3).