Theorem 1

A family $(x_i)$ of finite real numbers $\geqslant 0$ is summable in $\mathbf{R}$ if and only if the set of partial finite sums of the family is bounded above in $\mathbf{R}$. If so, the least upper bound of this set is the sum of the family $(x_i)$.

For each finite subset $H$ of the index set $I$, let $s_H = \sum_{i \in H} x_i$; since the $x_i$ are $\geqslant 0$, the relation $H \subset H'$ implies $s_H \leqslant s_{H'}$. In other words, the mapping $H \to s_H$ is an increasing function on the directed set $\mathfrak{F}(I)$ of finite subsets of $I$; therefore (§ 5, no. 2, Corollary to Theorem 2) it has a finite limit if and only if it is bounded above.