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?



Criterion for visuality of hyperbolic spaces
Let $x,y \in X$ be two points. If $X$ is bounded, there is nothing to prove. Otherwise, $X$ must have a boundary of cardinality at least two. So we can fix two distinct points $\zeta, \xi \in \partial X$, and a bi-infinite geodesic $\gamma$ between them. As $X$ is cobounded, up to translating $\gamma$ by an isometry, we may suppose without loss of generality that $y$ is at a controlled distance from $\gamma$. Because an ideal triangle we fix $\gamma \cup [x,\zeta) \cup [x,\xi)$ must be thin, it is not difficult to show that $y$ is at a controlled distance from either $[x,\zeta)$ or $[x,\xi)$. 


