Proper action

Given an manifold $X$ and a Lie group $G$ acting on $X$ , then the action is called proper if given a compact set $K \subseteq X$ then ${g \in G : g K \cap K \neq \emptyset}$ has compact closure in $G$ (alternative definition at @sharpe2000differential page 145). The meaning (not obvious) is that the orbits don't get too horrible. For example, the action of $Z$ on $S^{1}$ given by irrational rotations

n \cdot e^{i θ} = e^{i (θ + 2 π n α)}

with $0 < α < 1$ irrational is not a proper action.

Proposition
Given a Lie group $G$ and a closed subgroup $H$ then the right action of $H$ on $G$ is proper.
$◼$
(see @sharpe2000differential Proposition 2.2 page 145).