What do you call this method of proof?

Well normally when you are using n as an argument and showing it is true for all n in the natural numbers greater than k starting from some particular natural number k, you are doing a proof by induction. However, in your case it seems to me that you are relating n with an angle in some way. Since the angle is defined over the reals mod 2pi, you would need to be careful about trying to use induction to prove this. I would say try to use geometric and analysis arguments to prove your statements. You know how to relate angles to lengths using trigonometry, and so you have bounds on these lengths. You could show how the lengths change as a function of the angle and try to show what happens when the angle approaches some value.