Exercise 6, Appendix 0

Let $A$ be the algebra (over $\mathbf{R}$) of bounded real-valued functions defined on the interval $X = [1, +\infty[$ of $\mathbf{R}$, with the norm $\|f\| = \sup_{x \in X} |f(x)|$. For each integer $n > 0$, let $u_n$ be the function which is equal to $1/n$ for $n \leq x \leq n+1$ and equal to $0$ elsewhere. Show that the family $(1 + u_n)$ is multipliable in $A$ but that the series whose general term is $u_n$ is not absolutely summable in $A$.