I am going through some basic properties of $\delta$-hyperbolic spaces and groups and I am having some difficulties proving precisey some things that are anyway intuitively clear to me.
Let $G$ be a $\delta$-hyperbolic group and $H \subset G$ a $K$-quasiconvex subgroup, i.e. geodesics joining points of $H$ lie in the $K$-neighbourhood of $H$. I want to prove a similar statement for infinite geodesics starting at some element of $H$ and converging at infinity to some limit point of $H$.
More precisely let $\partial G$ denote the Gromov boundary of $G$ and $\Lambda (H)$ the limit set of $H$, i.e. the subset of $\partial G$ of accumulation points for sequences from $H$. Now take a geodesic ray $\gamma : [0,+\infty [ \to G$ such that $\gamma (0)=h\in H$ and $\gamma \to \xi \in \Lambda (H)$ at infinity. I want to prove that $\gamma$ lies in the $K$-neighbourhood of $H$.
What I understand: I can approximate $\gamma$ with finite-length geodesics $\gamma_n$ leaving $h$ and coming back to $H$ at some $h_n \in H$, where $\{ h_n \}$ are a sequence in $H$ converging at infinity to $\xi$. By $K$-quasiconvexity, $\gamma_n$ lies in the $K$-neighbourhood of $H$ for every $n$, so this should remain true passing to the limit $n\to \infty$.
To make this hand-waving precise, we need a notion of convergence for geodesic rays in $\overline{G}=G \cup \partial G$ (which of course we have), but I would like to see a more elementary proof of this fact, maybe something that builds a convenient geodesic polygon and then uses nothing more than the definitions of hyperbolicity and quasiconvexity.