I am trying to prove the following but so far did not succeed. Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side is contained in the $\delta$-neighbourhood of the union of the other two sides) and two (unit speed) geodesic rays $\gamma_1, \gamma_2$ originating from the same point $x$. If there is some $t$ such that $d(\gamma_1(t), \gamma_2(t)) \leq \delta$ then for any $0 \leq t' \leq t$: $d(\gamma_1(t'), \gamma_2(t')) \leq \delta$. Any hints or a proof?
2026-03-25 18:59:59.1774465199
Distance between geodesic rays
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