Theorem 3

Let $(x_i)_{i \in I}$ be a family of finite real numbers; then the following statements are equivalent:

a) The family $(x_i)$ is summable in $\mathbf{R}$. b) The family $(|x_i|)$ is summable in $\mathbf{R}$. c) The set of finite partial sums of the family $(x_i)$ is bounded in $\mathbf{R}$.

Let $I_1$ be the set of all $i \in I$ such that $x_i \geq 0$, and $I_2$ the set of all $i \in I$ such that $x_i < 0$. The family $(x_i)_{i \in I}$ *[resp. $(|x_i|)_{i \in I}$] is summable if and only if each of the families $(x_i)_{i \in I_1}$ and $(x_i)_{i \in I_2}$ *[resp. $(|x_i|)_{i \in I_1}$ and $(|x_i|)_{i \in I_2}$] is summable (Chapter III, § 5, no. 3, Propositions 2 and 3). Now it comes to the same thing to say that $(x_i)_{i \in I_1}$ is summable, or that $(|x_i|)_{i \in I_1}$ is summable, or that the set of finite partial sums of the family $(x_i)_{i \in I_1}$ is bounded (no. 1, Theorem 1); and the same is true with $I_1$ replaced by $I_2$. The theorem follows immediately.

Theorem 3 shows that the summability in $\mathbf{R}$ of a family of finite real numbers depends only on the summability of the family of their absolute values.

We recall (Chapter III, § 5, no. 5, Proposition 6) that if $(x_i)$ and $(y_i)$ are two summable families of finite real numbers, then the family $(x_i + y_i)$ is summable and $\sum (x_i + y_i) = \sum x_i + \sum y_i$. Moreover, if $(x_i)$ is a summable family of finite real numbers and $a$ is any finite real number, then the family $(ax_i)$ is summable in $\mathbf{R}$, and we have $\sum ax_i = a \cdot \sum x_i$.