Exercise 13, § 7

Let $(u_n)$ be a sequence of real numbers $\geq 0$ such that $\lim_{n \to \infty} u_n = 0$ and $\sum_{n=0}^\infty u_n = +\infty$. Let $(p_n)$ be a strictly increasing sequence of numbers $> 0$ such that $\lim_{n \to \infty} p_n = +\infty$. Show that there exists a permutation $\sigma$ of $\mathbf{N}$ such that for each $n \in \mathbf{N}$, $\sum_{k=0}^n u_{\sigma(n)} \leq p_n$.