Proposition 1

If the family $(x_i)$ is summable, then every neighbourhood of $o$ contains all the $x_i$ except for a finite subfamily (in other words, if $I$ is infinite, we have $\lim x_i = o$ with respect to the filter of complements of finite subsets of $I$).

This necessary condition for a family $(x_i)$ to be summable is by no means sufficient in general, even if $G$ is complete; we shall see many examples later (see Chapter IV, § 7).