Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every sequence have Cauchy subsequence). Let $G$ a compact subgroup of $H$. Then, $G$ is a topological group and the natural function $X\times G\rightarrow X$ given by $(g,x)\rightarrow g(x)$ is continuos and give us a continuos action of $G$ on $X$. Since that $G$ is a compact topological group, given $g\in G$ exists a sequence of positive integers $n_k\rightarrow \infty$ such that $g^{n_k}\rightarrow e$. in particular, the sequence of function $\{g^{n_k}\}$ converge to indentity map of $X$ uniformly over compact subsets of $X$.

**The question:**

Does Exists a subsequence of $\{g^{n_k}\}$ with converge uniformly?