"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." With the obvious application of Cauchy to exprdess the limit
After the second underline you make the statement: we have that for every there exists an integer , such that...
You do not know that such an n exists for an arbitrary theta. You are doing the same thing that I have been complaining about all along, you are assuming that in a bounded infinite collection of distinct points, every point is a limit point. In such a situation all that Bolzano Weirstrass guarantees is the existence of a limit point, not that every point is a limit point. If you can't grasp that, then this discussion is indeed futile.
(This post was particularly difficult because of the small edit box. Is there a way to enlarge it?)
EDIT: Whoops! Almost forgot. Thanks for taking the trouble to write out the above proof in more detail. Some of it was quite helpful