Definition:
Suppose $\mathcal{A}$ and $\mathcal{B}$ are two covers of $X.$ We say that $\mathcal{B}$ refines $\mathcal{A}$ if each member of $\mathcal{B}$ is contained in some member of $\mathcal{A}.$ We say that $\mathcal{B}$ strictly refines $\mathcal{A}$ if each member of $\mathcal{B}$ is a proper subset of some member of $\mathcal{A}.$ if $\mathcal{B}$ strictly refines $\mathcal{A},$ we write $\mathcal{A} < \mathcal{B}.$
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 $\epsilon.$
Could anyone help me in the solution, please?