Lemma 2

If $I$ is a finite subset of $\mathbf{N}$ such that $\varphi(I) < 1/c$, then

For since $\sigma_J \leq \sigma_I$ for every subset $J$ of $I$, we have from (2)

and since also $\sigma_I \leq c \cdot \sup_{J \subset I} ||s_J||$, from (1), it follows that

which leads to the result.

Now let $(e + u_n)$ be a multipliable sequence in $A$, whose product is a unit; by Theorem 1 there exists a finite subset $J_0$ of $\mathbf{N}$ such that, for each finite subset $H$ of $\mathbf{N}$ which does not meet $J_0$, we have

By Lemma 2, it follows that $\sum_{i \in H} ||u_i|| - 1$ for every finite subset $H$ of $\mathbf{N}$ which does not meet $J_0$, and hence (Chapter IV, § 7, no. 1, Theorem 1) the family $(||u_n||)$ is summable in $\mathbf{R}$.