Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not true that
(*) every two points in $\partial X$ can be connected by a geodesic in $X$.
However, one can perform the following trick, which allows one to reduce many questions to those spaces which actually satisfy (*).
Let $Y$ denote the metric ultrapower of $X$ with respect to a non-principal ultrafilter. Then $Y$ is also hyperbolic and there is a natural isometric embedding $X\to Y$, which extends to $\partial X\to \partial Y$. It is not difficult to show that any two points of $\partial X$ can be joined by a geodesic in $Y$.
My question is the following.
Are the details of this construction written in any paper or book?
This was a part of "hyperbolic folklore" for a long time and all proofs are straightforward, but I could not find any written detailed proof which can serve as a reference.