Exercise 7, § 5

Let $(x_{mn})$ be a double sequence of points of a Hausdorff commutative group $G$, satisfying the following conditions : (a) The series defined by the sequence $(x_{mn})_{n \in \mathbf{N}}$ is convergent for each $m \geq 0$; let $y_m$ be its sum. (b) If we put $r_{mn} = \sum_{p=n}^{\infty} x_{mp}$, then the series defined by the sequence $(r_{mn})_{m \in \mathbf{N}}$ is convergent for each $n \geq 0$; let $t_n$ be its sum.

Show that for each $n \geq 0$ the series defined by the sequence $(x_{mn})_{m \in \mathbf{N}}$ is convergent, and let $z_n$ be its sum. The series whose general term is $y_m$ has the same sum as the series whose general term is $z_n$ if and only if $t_n$ tends to $0$ as $n$ tends to infinity.