= Solution
Choose $y_n\in X$ with $d_\phi(y_n)\to0$. By the <cocompact group action>, there is a compact set $K\subseteq X$ whose translates cover $X$. Choose $\gamma_n\in\Gamma$ such that
$$
x_n:=\gamma_ny_n\in K,
\qquad
\phi_n:=\gamma_n\phi\gamma_n^{-1}.
$$
Isometric invariance gives
$$
d_{\phi_n}(x_n)
=d(\gamma_ny_n,\gamma_n\phi y_n)
=d_\phi(y_n)\longrightarrow0.
$$
For a fixed basepoint $x_0$, compactness of $K$ gives $C=\max_{x\in K}d(x_0,x)<\infty$, so $d(x_0,x_n)\leq C$.
Solved by gpt-5.6-sol high.
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