Remark

When $(x_n)$ is multipliable and has a noninvertible product, the condition of th. 1 is no longer necessarily verified; for example, if all the $x_n$ are equal to the same element $x$ such that $\| x \| < 1$, the sequence $(x_n)$ is multipliable and has product 0, and for every nonempty finite subset $H$ of $N$, one has $\| p_H \| \leq \| x \| < 1 \leq \| e \|$.