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I'm reading the geometric group Theory notes by Bowditch. Assume X is a complete, locally compact geodesic space. A group of isometries G acts on X properly discontinuously. Then X/G is Hausdorff, complete and locally compact. Suppose some orbit is cobounded. How can I show the quotient is compact? A metric space is compact iff it's complete and totally bounded. We already know the quotient is complete. So one approach is to show the quotient is totally bounded. I would appreciate if you can give me a hint on showing the quotient to be totally bounded or provide a hint on another approach.

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    What is the definition of cobounded.? All points is an orbit projects to a single point. Therefore "some orbit is cobounded" is a confusing statement. – 2017-01-23
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    You might be confused about the meaning of "cobounded". It is not a property of an individual orbit. It is a property of the action as a whole. For the action of $G$ on $X$ to be cobounded means that there exists a bounded set $B \subset X$ such that every orbit of the action has nonempty intersection with $B$. – 2017-01-24

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Hint: In a complete geodesic locally compact metric space closed balls are compact, this is a theorem due to Cohn-Vossen, see here. I do not know if your textbook covers this result but the linked Urs Lang's notes provided a self contained proof. Now, what do you know about continuous images of compact topological spaces?