Let $M = G/K$ be a homogeneous space. It is easy to show, that the left action of $G$ on itself by multiplication is a free and proper action.
My question is, if the induced action $$G \times G/K \to G/K, \ (g, h.K) \mapsto (gh).K$$
of $G$ on $G/K$ is also proper? I already showed, that if the action is proper, we necessarily have, that $K$ is compact. But is this condition sufficient, too?
Edit: An action $\phi \colon G \times M \to M$ is called proper, if the map $$G \times M \to M \times M, \quad (g,x) \mapsto (x, \phi(g,x))$$ is a proper map.
By "unbounded sequence" I mean one eventually escaping from any compact set. By bounded I mean contained in a compact set.
If it were not proper there would exist an unbounded sequence $g_n\in G$ and a bounded sequence $x_n$ such that $g_nx_n$ is bounded. Choose representatives $\gamma_n\in G$ for each $x_n$. Then since $G\rightarrow G/K$ is proper, $\gamma_n$ and $g_n\gamma_n$ are bounded sequences. This contradicts the fact you already proved about the action of $G$ on itself.