Let ($X,d$) be a compact metric space. Let {$x_n$} be a sequence in $X$ such that $d(x_{n+1},x_n) \rightarrow 0$ as $n \rightarrow \infty$. If set of limit points of the sequence is finite, then for any two limit points $a,b \in X$, show that $d(a, b) < \epsilon$ for all $\epsilon > 0$.
Note that my question is not just to prove that the sequence is convergent if number of limite points for the sequence is finite, but to prove the exact statement in the question. If I fix an $\epsilon > 0$, can I control the sequence to show that distance between any two limit points is less than $\epsilon$.