Let $(X,d)$ be a metric space.


Let $(x\_n)\_{n\in\mathbb N}$ be a geodesic ray: $d(x\_n,x\_m)=\vert n-m\vert$. 



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> 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 ?
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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.



Geodesic rays and horofunctions
**Yes**. Let $a\_n:=d(y,x\_n)-|n|$. By the triangle inequality, we see that $(a\_n)\_{n\ge0}$ is non-increasing and bounded. Therefore it has a finite limit $a$.


Now if $n\ge0$,
$d(y,x\_{n+1})-d(y,x\_n)=a\_{n+1}-a\_n+1\rightarrow a-a+1=1$.


