Properly Discontinuous Group Actions

Question

Let $G$ be a group acting on a topological space $X$. We say the action of $G$ on $X$ is properly discontinuous (or $G$ acts properly discontinuously on $X$) if one of the following two equivalent conditions is satisfied:

1. for every $x \in X$ and every compact subset $K$ of $X$ we have that the set $\{T \in G : T(x) \in K\}$ is finite.
2. for every compact subset $K$ of $X$ we have that the set $\{T \in G : T(K) \cap K \neq \phi \}$ is finite.

The question is how to prove the equivalence of these two conditions.