Let $X$ be a topological space, and let $G\subseteq Homeo(X)$ be a group of homeomorphisms of $X$ where $G$ acts properly discontinuously

Question

Let $X$ be a topological space, and let $G\subseteq \textrm{Homeo}(X)$ be a group of homeomorphisms of $X$.

a) Prove that if $G$ is finite, then $G$ acts properly discontinuously on $X$.

b) Suppose $X$ is compact and that $G$ acts properly discontinuously on $X$. Prove that $G$ is finite.

Any help is appreciated. I have included the definition below:

Definition: A group $G\subseteq Homeo(X)$ acts properly continuously on $X$ if for any compact set $C\subseteq X$, $|\{g\in G \mid C\cap gC \neq\emptyset\}| < \infty$

Answer 1

Item $a)$ follows immediately from the fact that if $G$ is finite, then so is each set $\{g\in G:C\cap gC\neq\emptyset\}$.

Item $b)$ follows by applying the fact that, since $X$ is compact, the set $\{g\in G:X\cap gX\neq\emptyset\}$ is finite. But then it is clear that $G=\{g\in G:X\cap gX\neq\emptyset\}$, so $G$ is finite.