But F(pi) + F(0) depends on n, rather than being a number, just this isn't explicitly spelled out. So all the proof shows is that the integral (1) converges to 0 rather quickly, as a function of n.

The claim of the proof is that I=F(0)+F(π) is simultaneously positive and arbitrarily small, which is not possible for a fixed number, but totally possible if you see I as a sequence indexed by n.