I am not sure how to proceed with this proof.

Say that a setKin a metric space is 'Tompac' if every closed cover ofKhas a finite subcover. show that such aKmust be a finite set.

(hint: any singleton set in a metric space is ...)

I know that any singleton set in a metric space is closed. my thinking on this is that a closed set contains the set and its limit points so if the set has a closed subcover say B[p, epsilon] then the set must be a subset of the cover but I am not sure how to show the set must be finite.