Let and be metric spaces and let be a dense subset of . Let and be continuous functions. Suppose that for all . Prove that .
Proof. Let . Then is a limit point of by the denseness of . Pick a decreasing sequence of positive real numbers . Then in each open ball , there is a point such that . Form these terms as a sequence . Fix . Since any open ball centered at contains at least one for some . So there is some such that . Then for , . So . Then . We know that for . Thus which implies . Thus .
Is this correct?