Exercise 9, § 9

Let G be a Lie group operating properly on a manifold X. Make one of the following two assumptions:

(i) $X/G$ is compact, and X is countable at infinity and of bounded dimension; (ii) X is a finite dimensional vector space on which G operates linearly.

Prove that the set of orbit types of elements of X is finite (treat the two cases simultaneously by induction on dim X, and use Prop. 6).