Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is equivalent to, having given $G$ the discrete topology and in the $X$ locally compact Hausdorff case, the map $G \times X \rightarrow X \times X$ given by $(g, x) \mapsto (x, gx)$ being proper (i.e. closed and preimage of compact sets is compact) plus the action being free.

I have managed to prove one direction, that is, if the action is proper and free with $G$ having the discrete topology then it is properly discontinuous. I'm having trouble though with the other direction. Here is an attempt: let's denote by $\rho : G \times X \rightarrow X \times X$ the map $\rho(g, x) = (x, gx)$. Suppose $K \subset X \times X$ is compact. We wish to show $\rho^{-1}(K)$ is compact. Let $(g_i, x_i)$ be a net in $\rho^{-1}(K)$. Then $\rho(g_i, x_i) = (x_i, g_i x_i)$ admits a convergent subnet, so passing to it we may assume $x_i \rightarrow x$ and $g_i x_i \rightarrow y$. Essentially we must now find a way to prove $g_i$ converges, but I can't seem to do this. Any hints?