Lemma 2

If $I$ is a finite subset of $\mathbf{N}$ such that $\varphi(I) < 1/c$, one has $\sigma_I \leq \frac{c \varphi(I)}{1 - c \varphi(I)}$.

Indeed, since $\sigma_J \leq \sigma_I$ for every subset $J$ of $I$, one has, after (2),

since, by virtue of (1), one has $\sigma_I \leq c \cdot \sup_{J \subset I} \| s_J \|$, it follows that $\sigma_I \leq c \varphi(I)(1 + \sigma_I)$, whence the lemma.

That being so, let $(e + u_n)$ be a multipliable sequence in $A$, whose product is invertible; according to th. 1 (IX, p. 79), there exists a finite subset $J_0$ of $\mathbf{N}$ such that, for every finite subset $H$ of $\mathbf{N}$ not meeting $J_0$, one has

By Lemma 2, one deduces $\sum_{i \in H} \| u_i \| \leq 1$ for every finite subset $H$ of $\mathbf{N}$ not meeting $J_0$, which implies that the family $(\| u_n \|)$ is summable in $\mathbf{R}$ (IV, p. 32, th. 1).