Exercise 2, § 3

Let $(x_m)_{m \in \mathbf{N}}$ be an infinite sequence of points of $\mathbf{R}^n$ such that $\lim_{m \to \infty} x_m = 0$. For each finite subset $H$ of $\mathbf{N}$, put $s_H = \sum_{m \in H} x_m$. Show that there are two possibilities: either

(i) $\lim_{\mathscr{F}} \|S_H\| = + \infty$, where $\mathscr{F}$ is the directed set of all finite subsets of $\mathbf{N}$; or else:

(ii) there exist permutations $\sigma$ of $\mathbf{N}$ such that the series whose general term is $x_{\sigma(m)}$ is convergent in $\mathbf{R}^n$; in this case the set $A$ of sums $\sum_{m=0}^\infty x_{\sigma(m)}$ of these series, for all permutations $\sigma$ with this property, is an affine linear variety in $\mathbf{R}^n$. [Show first, with the help of Exercise 1 b), that every cluster point (with respect to $\mathscr{F}$) of the mapping $H \to S_H$ is the sum of a convergent series $(x_{\sigma(m)})$ for a suitable permutation $\sigma$. Use Chapter III, § 5, Exercise 3 to prove that the set $A$, if not empty, is a coset of a closed subgroup of $\mathbf{R}^n$. Show finally that $A$ is connected by means of Exercise 1 c) and Chapter II, § 4, Exercise 15.]