Exercise 4, § 8

Let $(u_n)$ be a summable sequence of numbers > 0 in $\mathbf{R}$ which satisfies the following conditions : $u_{n+1} \leq u_n$ and $u_n \leq \sum_{k=1}^{\infty} u_{n+k}$ for all $n \geq 0$. Show that for every number $a$ such that $0 < a \leq s = \sum_{n=0}^{\infty} u_n$, there exists a subset I of $\mathbf{N}$ such that $a = \sum_{n \in I} u_n$. These conditions are satisfied in particular if $u_{n+1} \leq u_n \leq 2u_{n+1}$ for all $n$. Consider the case of dyadic expansions.