I don't see how this question reduces to the density of the rationals in [0,1] - surely not every dense set in [0,1] is , right? I mean, for example, is not rational..So I have a countably infinite number of points on the circle. Are they dense?
Somehow this has to get to the equivalent question, are the rational points on [0,1] dense? Yes, because every real point of the interval is a limit point of the rational numbers on the interval.
Fernando's proof essentially seals the deal. What do you not understand about it?An infinite number of points on the circle must indeed have a limit point, but is every point hit as per above a limit point, in terms I could understand? In other words, given any point on the circle, are there points constructed as per above on the circle within an arc length epsilon?
EDIT: So I have the question, given any point on the unit circle, how many times do I have to go around to get to within a distance epsilon? Getting close.