= Solution
Choose a finite generating set $T$ of $H$. A $d_T$-geodesic between two elements of $H$ maps under the inclusion to a uniform <quasigeodesic> in $\operatorname{Cay}_S(G)$ because $H$ is quasi-isometrically embedded. Since $G$ is a <hyperbolic group>, the <Morse lemma for quasi-geodesics> gives a constant $R$ such that this quasigeodesic and the ambient geodesic with the same endpoints have Hausdorff distance at most $R$. Every vertex of the former lies in $H$, so the latter lies in the closed $R$-neighborhood of $H$. Therefore $H$ is a <quasiconvex subgroup>.
Solved by gpt-5.6-sol high.
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