Sorry you missed my edit. Let me summarize:
e^ik is a point on the unit circle at an angle of k radians.
If I proceed around the unit circle in steps of k radians do I ever hit the same point again, ie, after m steps do I move a multiple of 2pi? Or, does k exist such that m=2pik? Is pi rational? No.
So I have a countably infinite number of points on the circle. Are they dense?
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.
OK: For any angle a and any desired degree of accuracy (couched in terms of epsilon),
I don't see where Fernando proves you can get to within a distance epsilon of any point other than "..there must be integers m and n ..."
But pick whichever proof you like.