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 ?
2026-03-27 19:53:12.1774641192
Arzela-Ascoli in proper metric space.
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