Lemma (i)

If $(a_i)_{1 \leq i \leq p}$ is a finite sequence of numbers $> 0$, then

(ii) If also $a_i < 1$ for each $i$, then

These inequalities are clear if $p = 1$, and are proved by induction on $p$. If

we have Again, if we have

Having proved the lemma, we note that if the family $(1 + u_i)$ is multipliable then so are the families $(1 + u_i^+)$ and $(1 - u_i^-)$, since $\mathbf{R}^*$ is a complete group (Chapter III, § 5, no. 3, Proposition 2); and conversely, if the families $(1 + u_i^+)$ and $(1 - u_i^-)$ are multipliable, then so is $(1 + u_i)$ (Chapter III, § 5, no. 3, Proposition 3). We need therefore consider only the cases in which all the $u_i$ are $\geq 0$ and in which they are all $\geq 0$.

Suppose first that $u_i \geq 0$ for all $i$. If the family $(1 + u_i)$ is multipliable, then for each $\varepsilon > 0$ there is a finite subset $J$ of the index set in $I$ such that, for each finite subset $H$ of $I$ disjoint from $J$, we have $1 \leq \prod_{i \in H} (1 + u_i) \leq 1 + \varepsilon$; by (2) it follows that $\sum_{i \in H} u_i \leq \varepsilon$, which shows that $(u_i)$ is summable in $\mathbf{R}$ by virtue of Cauchy’s criterion (Chapter III, § 5, no. 2, Theorem 1).

Conversely, suppose that $(u_i)$ is summable in $\mathbf{R}$. For each $\varepsilon$ such that $0 < \varepsilon < 1$, there is a finite subset $J$ of $I$ such that, for each finite subset $H$ of $I$ disjoint from $J$, we have $0 \leq \sum_{i \in H} u_i \leq \varepsilon$. By (3), we have therefore $\prod_{i \in H} (1 - u_i) \geq 1 - \varepsilon$; but since $1 + u \leq \frac{1}{1 - u}$ for every number $u$ such that $0 \leq u < 1$, it follows that

and this shows that $(1 + u_i)$ is multipliable (Cauchy’s criterion).

The proof is similar when all the $u_i$ are $\leq 0$. To show that $(u_i)$ is summable if $(1 + u_i)$ is multipliable, use formula (2) and the inequality $1 - u \leq \frac{1}{1 + u}$ ($0 \leq u < 1$); to show that $(1 + u_i)$ is multipliable if $(u_i)$ is summable, use formula (3).

In Chapter V (§ 4), the topological study of the group $\mathbf{R}^*$ will enable us to give another criterion for multipliability of a family in $\mathbf{R}^*$ with the help of the logarithmic function; in a later volume we shall establish the equivalence of this criterion with the one above, using the differential properties of the logarithm.