# Are all Gromov hyperbolic spaces proper metric spaces

Here a proper metric space is a metric space such that all closed balls are compact. My question is are all Gromov hyperbolic spaces proper metric spaces? I only know the rudimentary definitions of these things which is why I am asking, I didn't see it referenced anywhere and I was curious.

Even simpler, any infinite set $X$ endowed with the discrete metric, ie. $$(x,y) \mapsto \left\{ \begin{array}{l} 0 & \text{if x=y} \\ 1 & \text{otherwise} \end{array} \right.,$$ provides an example of a non proper Gromov hyperbolic space.