Exercise 6, Appendix 2

Let $A$ be the algebra (over $\mathbf{R}$) of bounded numerical mappings defined on the interval $E = [1, + \infty[$ of $\mathbf{R}$, normed by the norm $\|f\| = \sup_{x \in E} |f(x)|$. For every integer $n > 0$, let $u_n$ be the function equal to $1/n$ for $n \leq x \leq n+1$, and to 0 at the other points of $E$. Now prove that the family $(1 + u_n)$ is multipliable in $A$, but that the series of general term $u_n$ is not absolutely summable in $A$.