Let A be a bounded subset of , which contains infinitely many different points. Prove that there exists at least one point such that all open intervals around contains infinitely many different points from A.
In this case, I would try to construct a limit point.
First, can be contained in a closed interval . Split in half. One of the parts must contain infinitely many points from . Take that one and call it . Then, split in half. One of those parts must contain infinitely many points from , so take that one and call it . Continue this procedure indefinitely to receive a sequence of closed intervals such that whose sizes are , respectively.
Now, by the Nested Interval theorem, contains a single point, and that point is the limit point we are looking for.