Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

Question

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

I am working through some notes on Geometric Group Theory and I am having a hard time with this problem. Some help would be great.

A topological group action $μ : G$ acts on $X$ is called proper if for every compact subsets $K_1,K_2 ⊂ X,$ the set is $\{g \in G | g(K_1)\cap K_2 \neq \emptyset \}$ contained in $G$ is compact.