In the book by Bridson and Haefliger http://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf page 145. lemma 8.28 To prove part 2 of the lemma that is The natural map for $G_{x_0}\rightarrow Ends(X)$ is surjective. They use Arzela-Ascoli theorem for which the the image space need to be compact. But only assume X is a proper metric space. For a proper ray $r:[0,\infty)\rightarrow X$ the image might not be compact. Can anyone please explain ?
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I do not know the context, but you can use uniform convergence on compacts for such maps together with diagonal subsequence arguments. – Moishe Kohan Jan 22 '15 at 04:09
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Thanks for the comment. I missed the fact that convergence on compact set. – GGT Jan 22 '15 at 09:54