Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ is cobounded if there is $r\ge 0$ such that for every $y\in X$, there is some $q\in Q$ such that $d(y,q) \le r$.

I am trying to prove that the quotient space $X/G$ is compact if and only if every orbit of $G$ is cobounded in $X$.

I'd very much appreciate if someone could let me know whether or not the following is valid. In particular, page 19 of <https://www.math.ucdavis.edu/~kapovich/280-2009/bhb-ggtcourse.pdf> suggests that I need the additional assumption that the action of $G$ is properly discontinuous. However, my proof did not use this fact.

Proof (?): 

(1) $\to$ (2): We may give $X/G$ a metric. Define $$d\_{Q} (G\cdot x, G\cdot y) = \min\{d(x, g\cdot y) : g\in G\}$$

It is straightforward to check this metric induces the quotient topology. Choose some orbit $G\cdot x \in X/G$. There is $r$ such that $X/G \subset N(G\cdot x, r)$. For any $y\in X$, it follows $d\_{Q} (G\cdot y, G\cdot x) \le r$, hence $d(y, g\cdot x) \le r$ for some $g\in G$. Thus $G\cdot x$ is cobounded.

(2) $\to$ (1): Again, choose $G\cdot x \in X/G$. Assume $G\cdot x$ is $r$-dense in $X$. Then, for any $y\in X$, there is $g\in G$ such that $d(x,g\cdot y) \le r$. By Hopf-Rinow, we know $N(x, r)$ is compact. The projection map $q: X \to X/G$ given by $x \mapsto G\cdot x$ is continuous. Hence $q(N(x,r)) = X/G$ is compact.