= Solution
Let $h\in H$ and let
$$
1=x_0,x_1,\ldots,x_N=h
$$
be a geodesic in the <Cayley graph> $\operatorname{Cay}_S(G)$. By $K$-quasiconvexity choose $h_i\in H$ with $d_S(x_i,h_i)\leq K$, taking $h_0=1$ and $h_N=h$. Then
$$
|h_i^{-1}h_{i+1}|_S
\leq d_S(h_i,x_i)+1+d_S(x_{i+1},h_{i+1})
\leq2K+1.
$$
The elements $h_i^{-1}h_{i+1}$ telescope to $h$, so the finite set
$$
U=H\cap\{g:|g|_S\leq2K+1\}
$$
generates $H$.
Moreover $|h|_U\leq|h|_S$, while $|h|_S\leq(2K+1)|h|_U$. Thus the inclusion $(H,d_U)\to(G,d_S)$ is a <quasi-isometric embedding>, and $H$ is quasi-isometrically embedded.
Solved by gpt-5.6-sol high.
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