Exercise 5, § 5

Let $(x_n)$ be a sequence of points of a Hausdorff commutative group. If, for every infinite subset $I$ of $\mathbf{N}$, the series defined by the sequence $(x_n)_{n \in I}$ is convergent, then the sequence $(x_n)$ is summable (argue by reductio ad absurdum as in the proof of Proposition 9).