I am currently studying the visual boundary $\partial X$ of a (complete) CAT(0) space $X$, firstly introduced with the cone topology as done in II.8 in Bridson-Haeflinger: "Metric spaces of non-positive curvature". Over there, it is simply stated that $\bar{X}$ is compact if $X$ is proper.
Since I had no idea why, I looked it up and found that the visual compactification is homeomorphic to the horofunction compactification as done in Ballmann his book "Manifolds of Nonpositive Curvature", and apply then an Arzela-Ascoli argument for the latter compactification to get the statement above.
However, I wonder, is there a more direct approach which avoids the horofunction compactification? If so, can you give a few hints on how to do so?