Exercise 8, § 4

Let $G_0$ be a non-discrete Hausdorff topological group in which every compact subset is finite ($\S$ 2, Exercise 23). Let $G$ be the group $G_0$ with the discrete topology. Show that $G$ operates continuously but not properly on $G_0$ by left translation, and is such that $P(K, L)$ is compact for all compact subsets $K, L$ of $G_0$ (Theorem 1).