Exercise 6, § 5

a) Let $\sigma$ be a permutation of $\mathbf{N}$ and let $\varphi(n)$ be the smallest number of intervals of $\mathbf{N}$ whose union is $\sigma([0, n])$. Suppose that $\varphi(n)$ is bounded as $n$ runs through $\mathbf{N}$. Then if $(u_n)$ is any convergent series whose terms belong to a Hausdorff commutative group $G$, the series $(u_{\sigma(n)})$ is convergent and has the same sum as $(u_n)$.

\* b) Suppose that $\varphi(n)$ is not bounded in $\mathbf{N}$. Construct a series $(u_n)$ whose terms belong to the additive group $\mathbf{R}$ and which converges in $\mathbf{R}$ but is such that the series $(u_{\sigma(n)})$ does not converge. [Consider a strictly increasing sequence $(m_k)$ of integers, defined by induction on $k$, satisfying the following conditions : (i) if $[0, n_k]$ is the largest interval of $\mathbf{N}$ with left-hand extremity $0$ which is contained in $\sigma([0, m_k])$, then

(ii) $\varphi(m_k) \geq k + 1$. Then define $u_n$ appropriately for $n_k < n < n_{k+1}$. Generalize to the case of a complete Hausdorff commutative group $G$ which is generated by every neighbourhood of $0$. \*