Here is the problem:
Let $X$ be a compact metric space that is totally disconnected, and let $\epsilon > 0.$
(a) Show that $X$ has a finite cover $\mathcal{A}$ clopen sets with diameter at most $\epsilon.$
My trial.
With the help of many people here on this site I was able to prove that:
If $X$ is a compact metric space that is totally disconnected, then for each $r > 0$ and each $x \in X,$ there is a clopen set $U$ such that $x \in U$ and $U \subseteq B_{r}(x).$
I feel like this may help me in the proof of the first question but I do not know how, could anyone clarify this for me, please?
Also, I received hints here finite cover of clopen sets. but still, I am unable to write the solution thoroughly. Any help will be appreciated.