I'm having trouble with understanding one detail in this proof. I'd greatly appreciate if anyone can help.

Let be a continuous function. Suppose exists and finite. Then is uniformly continuous on .

Proof: Let . Since as , there exists st when we have . We know is uniformly continuous on since is compact and is continuous. Hence, there exists st for every , . Choose . Then any two points satisfying are either both in or both in . I understand the rest of the proof. The important detail that I don't get is this lineI don't see why must be both in or both in . I guess it must have something to do with the distance between them less than delta, but I don't see it. I also don't see why we need the distance less than .Choose . Then any two points satisfying are either both in or both in