For a geodesic Gromov Hyperbolic metric space X is it true that there exists $C>0$ such that any two bi-infinite geodesic with same end points at boundary stays within $C$-neighbouhood of each other?
I know that geodesic rays going to same point in the boundary stays within a constant distance of each other after a finite time .
But I could not find an argument for to prove the prior statement, neither was I able to show find a counter example.
I shall be grateful if someone could comment on or help me in this regard .