Think of a collection of integrals, one for each n, rather than a function depending on multiple variables. The contradiction is that the number F(pi) + F(0) is both a positive integer and arbitrarily small.
There is no contradiction until n is large enough that F(pi) + F(0) is bounded below 1, hence the statement at the start: we'll pick a large enough value of n later.
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.
> 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.
This is not possible because I_n must be an integer for every n.
There is no contradiction until n is large enough that F(pi) + F(0) is bounded below 1, hence the statement at the start: we'll pick a large enough value of n later.