Remark

When G is written multiplicatively, Cauchy’s criterion takes the following form : for the family $(x_i)_{i \in I}$ to be multipliable it is necessary that, for each neighbourhood V of the identity element, there should exist a finite subset $J_0$ of I such that, for each finite subset K of I which does not meet $J_0$, we have $\prod_{i \in K} x_i \in V$; and this condition is sufficient provided that G is complete. We deduce that if I is infinite and $(x_i)$ is multipliable, then $\lim x_i = 1$ with respect to the filter of complements of finite subsets of I; if in addition the identity element has a countable fundamental system of neighbourhoods, then the set of indices i such that $x_i \neq 1$ is countable.