Let $(X,d)$ be a CAT($-1$) space. Let $c_1,c_2 : [0,+\infty) \rightarrow X$ be two asymptotic geodesic rays, i.e. two geodesic rays such that $\exists M > 0$, $\forall t \in [0,+\infty) \rightarrow X$, $d(c_1(t),c_2(t)) \leq M$.
It appears to be "well known" that there exists $u \in \mathbb{R}$ such that $\lim_{t \to \infty} d(c_1(t),c_2(t + u)) = 0$.
I know that this is true in the hyperbolic plane, but I cannot prove it for a general CAT(-1) space. Can you help me ?
EDIT : I have a guess on what $u$ should be. Let $b_t(x) := d(x,c_1(t)) - t$. Then $b_t(x)$ is decreasing, bounded below (by the triangular inequality) so it has a limit $b(x)$. I have serious reasons to believe that the right $u$ is $b(c_2(0))$.