Proposition 3

Let $(x_i)_{i \in I}$ be a family of points of a group $G$, and let $(I_\lambda)_{\lambda \in L}$ be a finite partition of $I$. If each of the subfamilies $(x_i)_{i \in I_\lambda}$ is summable, then the family $(x_i)_{i \in I}$ is summable and the formula (1) is valid.

It is enough to prove the proposition when $L = (1,2)$; having done so, we then proceed by induction on the number of elements of $L$. Let $s_1 = \sum_{i \in I_1} x_i$ and $s_2 = \sum_{i \in I_2} x_i$. For each neighbourhood $V$ of the origin, there is a finite subset $J_1$ (resp. $J_2$) of $I_1$ (resp. $I_2$) such that, for each finite subset $H_1$ (resp. $H_2$) of $I_1$ (resp. $I_2$) containing $J_1$ (resp. $J_2$), we have $\sum_{i \in H_1} x_i \in S_1 + V$ (resp. $\sum_{i \in H_2} x_i \in s_2 + V$). If we put $J_0 = J_1 \cup J_2$, it follows that, for each finite subset $H$ of $I$ which contains $J_0$, we have $\sum_{i \in H} x_i \in S_1 + s_2 + V + V$, and the result follows.