We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-geodesic rays up to finite Hausdorff. To show that $\partial X\to \partial_q X$ is surjective (in fact bijective) in any proper hyperbolic geodeisc space $X$, it suffices to show that quasi-geodeisc rays are closed to geodesic rays in $X$.
It follows from Bridson-Haefliger III.H.3.10 that let $p\in X$ and $c:[0,\infty)\to X$ a quasi-geodesic ray in $X$. We define $c_n$ to be the sequence of geodesics connecting $p$ and $c(n)$. Apply Arzela-Ascoli by properness of $X$, we have a subsequence $c_{n_i}$ converges to a geodesic ray $c_{\infty}$ and we conclude by stability of quasi-geodesics. My question is how properness of $X$ implies the hypothesis of Arzela-Ascoli i.e. $c_n$ is equicontinuous and bounded. It's clear for geodesics but not geodesic rays for me since the domain is infinite.
Thanks for your help!