so if a sequence in X has a finite range, does it have to be in one of the closed sets? if it is then i can take away finitely many terms so that the subsequence will converge. if the sequence has an infinite range then would it be contained in one of the open sets? based on that i don't know if it has a convergent subsequence. i'm still unsure on how to continue. is there still something i may be missing?
