Action of a Closed Lie subgroup

Question

Suppose $G$ is a Lie group acting properly on a manifold $P$.

Does it imply the action of any closed subgroup $H$ on $P$ is also proper?

Answer 1

The action of $G$ on $P$ is proper if and only if for every compact subset $C$ of $M$, $G_C=\{g:g(C)\cap C\neq \phi\}$ is relatively compact. If $H$ is closed, $H_C$ is a subset of $G_C$, its closure is contained in the closure of $G_C$ so it is compact, this closure is contained in $H$ since $H$ is closed, therefore the action of $H$ is proper.