Exercise 4, § 5

With the notation of Exercise 3, take $G$ to be the discrete additive group $\mathbf{Z}$ of rational integers and $I$ to be the set $\mathbf{N}$. Give an example of a non-summable sequence $(x_n)$ for which the set $A$ consists of the single point $o$ (choose the $x_n$ so that every finite partial sum, all of whose terms have indices $\geq m$, is an integral multiple of $m$).