Exercise 14, § 7

Let $(u_n)$ be a sequence of finite real numbers such that the series whose general term is $u_n$ is convergent but not absolutely convergent; let $s = \sum_{n=0}^\infty u_n$. Show that, for each number $s' \geq s$, there exists a permutation $\sigma$ of $\mathbf{N}$ such that $\sigma(n) = n$ for those indices $n$ such that $u_n \geq 0$, and such that $\sum_{n=0}^{\infty} u_{\sigma(n)} = s'$. [Show by induction on $m$ that there is a permutation $\sigma_m$ of $\mathbf{N}$ such that $\sigma_m(k) = k$ for all $k$ for which $u_k \geq 0$ and such that, if $u_k^{(m)}$ denotes $u_{\sigma_m(k)}$, there is an integer $p_m$ with the property that $\left| s' - \sum_{i=0}^k u_i^{(m)} \right| \leq 1/m$ for all $k \geq p_m$. Moreover, $\sigma_{m+1}$ is such that $\sigma_{m+1}(k) = \sigma_m(k)$ for all $k$ such that $\sigma_m(k) < p_m$ and all $k$ such that $u_k \leq -1/m$.]