Let $(X,d)$ be a metric space.
Let $(x_n)_{n\in\mathbb N}$ be a geodesic ray: $d(x_n,x_m)=\vert n-m\vert$.
Is it true that, for all $y\in X$, the sequence $(d(x_{n+1},y)-d(x_n,y))$ converges to $1$ as $n$ goes to infinity ?
I am particularly interested in the case of $\delta$-hyperbolic spaces. A positive answer to the above question would imply that any geodesic ray converges to a Busemann function (or horofunction).
More generally, is anything known about Busemann functions on hyperbolic spaces ? In particular, how do the Busemann compactification relates to the visual boundary ? These two boundaries are the same for CAT(0) spaces, but need not be in general, as shown by the example $\mathbb Z\times \mathbb Z/2\mathbb Z$, with obvious generating set.