Proposition 2

In a complete group G, every subfamily of a summable family is summable.

For if Cauchy’s criterion is satisfied by a family $(x_i)_{i \in I}$, then it is trivially satisfied by every subfamily.

If $(x_i)_{i \in I}$ is summable, it follows therefore that the sum $\sum_{i \in J} x_i$ is defined for every subset J of I, finite or not : it is again called the partial sum of the family $(x_i)$, corresponding to the subset J of the index set. The set of partial sums of a summable family is evidently contained in the closure of the set of finite partial sums.