Exercise 15, § 7

Let $(u_n)$ be a sequence of finite real numbers such that $\lim_{n \to \infty} u_n = 0$ and $\sum_n u_n^+ = \sum_n u_n^- = +\infty$. If $a$ and $b$ are any two real numbers (finite or not) such that $a \leq b$, show that there is a permutation $\sigma$ of $\mathbf{N}$ such that, if we put $s_n = \sum_{k=0}^n u_{\sigma(k)}$, we have $\liminf_{n \to \infty} s_n = a$ and $\limsup_{n \to \infty} s_n = b$. Show that the set of cluster points of the sequence $(s_n)$ is then the interval $[a, b]$.