Let $G\leq Sym(X)$ ($Sym(X)$ is the group of permutations of $X$, with the product topology) be a closed subgroup, such that every $G$-orbit is finite.
I want to show that $G$ is compact (I've already proved the converse, and I read somewhere that the two were equivalent); but I've got no idea how to do it. I've tried several characterizations, but each time I get stuck with essentially the same problem: I never get a single $x$ for which I could use the hypothesis; in other words the most obvious things don't work.
I would like it very much if you could provide a hint or so; but not a complete solution (if possible; sometimes a hint amounts to a solution)