Definition: G : topological group. A continuous action $\alpha : G \times M \rightarrow M$ is called a proper action if the following map is proper:
$(\alpha; \text{id}) : G \times M \rightarrow M \times M: (g; m) \mapsto (\alpha(g, m), m) $
Suppose that $G$ is compact and $M$ is Hausdorff. I am trying to prove that a continuous action of $G$ on $M$ is proper.
where
Definition: Let $f : X \rightarrow Y$ be a continuous map. $X,Y$ Topological spaces. $f$ is called proper if $f^{-1}(K)$ is compact for every compact $K \subseteq Y$.
and the following proposition might be useful
Proposition: If $Y$ is a metric space and $f: X \rightarrow Y$ is continuous and proper, $f$ is closed
My try: Let $A:=(\alpha,id)$ and $C$ be compact in $M\times M$ I want to prove that $A^{-1}(C)$ is compact. Because $C$ is compact in $M\times M$ that is Hausdorff, then $C$ is closed there and by continuity of $A$, $A^{-1}(C)$ is closed in $G \times M$ .I am stuck here maybe if I can construct some compact set where this last set is included I could use that a closed set in a compact space is compact.
Any help is appreciated