Proof That a Compact Set is Countably Compact
A compact set is countably compact.
Supposeis compact. Let be a subset ofwith no accumulation points in
Each pointbelongs to some open setwhich contains at most one point ofConsider the family of sets
Henceis an open cover ofSinceis compact, a finite subcoverexists with
Since eachcontains at most one point ofis finite. Therefore every infinite subset ofcontains an accumulation point inso thatis countably compact.