Descartes' proved that adding the angular deficits of all such vertexes, no matter their number, yields a constant number, 720 degrees. Ergo Sigma (360 - v) over all N = 720. This proves the limit at each vertex is never zero, as every vertex contributes some tiny "tax" or "tithe" to the invariant constant 720. 720/N > 0. |360 - v| > 0 even as N -> infinity.
A contradiction would involve two statements, but here there is only one, that the deficit (360 - v) approaches zero as N increases without bound. That a function has a limit at a point doesn't mean that the function exists at that point. The limit of 1/x as x increases without bound is 0, but 0 isn't in the range of 1/x, nor is infinity in the domain, nor can it even be in any domain.