Exercise 16, § 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. Show that there exists a permutation $\sigma$ of $\mathbf{N}$ satisfying the condition of Chapter III, § 5, Exercise 6 a) but such that the series whose general term is $u_{\sigma^{-1}(n)}$ is not convergent. [Let $h, k, m$ be three integers with the following properties : if $p_1, \ldots, p_r$ are the integers $n$ such that $h \leq n < h + m$ and $u_n \geq 0$, then $u_{p_1} + \cdots + u_{p_r} > 2$ and $h + m < k - r$; moreover if $\mu, \nu$ are any integers such that $h + m \leq \mu \leq \nu$, then $\left| \sum_{n=\mu}^{\nu} u_n \right| \leq 1$. Put $s = m - r$ and let $q_1, \ldots, q_s$ denote the integers $n$ such that $h \leq n < h + m$ and $u_n < 0$. Consider the permutation $\pi$ of the interval $[h, k + s]$ of $\mathbf{N}$ such that $\pi(p_i) = k - i + 1$ for $1 \leq i \leq r$, $\pi(q_j) = k + j$ for $1 \leq j \leq s$, $\pi(k + j) = h + s - j$ for $1 \leq j \leq s$, $\pi(k - i + 1) = h + s + i - 1$ for $1 \leq i \leq r$ and finally $\pi(n) = n$ for $h + m \leq n \leq k - r$; show that we have