Exercise 17, § 2

Let $G$ be the additive group of mappings $n \to f(n)$ of $\mathbf{N}$ into $\mathbf{Q}$ such that $\lim_{n \to \infty} f(n)$ exists in $\mathbf{R}$. For each $\alpha > 0$, let $V_\alpha$ be the set of $f \in G$ such that $|f(n)| < \alpha$ for all $n \in \mathbf{N}$. Show that the $V_\alpha$ satisfy axioms (GV_I) and (GV_{II}) and that with the topology defined by the $V_\alpha$, the group $G$ is Hausdorff and totally disconnected. Let $H$ be the subgroup of all $f \in G$ such that $\lim_{n \to \infty} f(n) = 0$. Show that $H$ is closed and that $G/H$ is isomorphic to $\mathbf{R}$ and therefore connected. \*