I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."

Let $X$ be a proper geodesic hyperbolic space. $X$ is called cobounded if there exists $R>0$ such that for all $x,y \in X$ there exists an isometry $f$ of $X$ such that $d(f(x), y) < R$. It is called visual, if for some (hence any) basepoint $o$ of $X$ there exists $R>0$ such that every $x\in X$ has distance at most $R$ from some geodesic ray emanating from o.

From the formulation it sounds that the sentence above should be an obvious fact, but I don't see how one should approach this. Can anyone provide a hint, how one would even get started?