Exercise 11, § 4

a) Let $G$ be a locally compact group generated by a compact neighborhood of $e$. Let $\varphi$ be a non-surjective continuous endomorphism of $G$ belonging to the closure, for the topology of compact convergence, of the group $\mathscr{G}$ of (bicontinuous) automorphisms of $G$. Then $\lim_{\psi \in \mathscr{G}, \psi \to \varphi} \mod \psi = 0$. (Let $K$ be a compact neighborhood of $e$ that generates $G$. Let $\mu$ be a left Haar measure on $G$. If $\mu(\varphi(K)) > 0$, then $\varphi(K) \cdot \varphi(K)^{-1}$ is a neighborhood of $e$ in $G$, therefore $\varphi(G)$ is an open subgroup of $G$; for $\psi \in \mathscr{G}$ sufficiently near $\varphi$, one has $\psi(K) \subset \varphi(G)$, therefore $\psi(G) \neq G$, which is absurd. Thus $\mu(\varphi(K)) = 0$. As $\psi \in \mathscr{G}$ tends to $\varphi$, $\mu(\psi(K))$ tends to 0.) b) Let $G$ be a free abelian group, a direct sum $G_1 \oplus G_2 \oplus \cdots$, where each $G_i$ is isomorphic to $\mathbf{Z}$. Consider $G$ as being discrete. Let $\varphi$ be the non-surjective endomorphism $(x_1, x_2, \ldots) \mapsto (0, x_1, x_2, \ldots)$ of $G$. Then $\varphi$ is the limit, for the topology of compact convergence, of automorphisms of $G$, and every automorphism of $G$ has modulus 1.