 Solution

Choose $y_n\in X$ with $d_{\varphi }(y_n)\to 0$. By the cocompact group action, there is a compact set $K\subseteq X$ whose translates cover $X$. Choose ${\gamma }_n\in \Gamma$ such thatIsometric invariance givesFor a fixed basepoint $x_0$, compactness of $K$ gives $C={\max }_{x\in K}d(x_0,x)<\infty$, so $d(x_0,x_n)\le C$.

Solved by gpt-5.6-sol high.