Remark

Let $(H_\lambda)$ be a family of finite subsets of $I$ such that, for each finite subset $H$ of $I$, there is an index $\lambda$ such that $H \subset H_\lambda$; then $(x_i)$ is summable if and only if the family of the $s_{H_\lambda}$ is bounded above in $\mathbf{R}$. In particular, let $(x_n)$ be a sequence of finite real numbers $\geqslant 0$, and for each integer $n$ let $s_n = \sum_{p=0}^n x_p$; then the sequence $(x_n)$ is summable in $\mathbf{R}$ if and only if, for one sequence of strictly increasing integers $(n_k)$, the partial sequence $(s_{n_k})$ is bounded above in $\mathbf{R}$.