real analysis: finite cover of clopen sets with maximum diameter $ epsilon. $


Here is the problem:

Leave $ X $ be a compact metric space that is totally disconnected and leave $ epsilon> 0. $

(a) Show that $ X $ it has a finite cover $ mathcal {A} $ clopen sets with a maximum diameter $ epsilon. $

My judgment

With the help of many people here on this site, I was able to demonstrate that:

Yes $ X $ is a compact metric space that is totally disconnected, then for each $ r> 0 $ and every $ x in X, $ there is a clopen set $ U $ such that $ x in U $ Y $ U subseteq B_ {r} (x). $

I feel this can help me in testing the first question, but I don't know how, could someone please clarify this for me?

Also, I got hints here from a finite deck of clopen sets. but still, I can't write the solution thoroughly. Any help would be appreciated.