Proof that Pi is Irrational

Suppose \(\pi = a/b\). Define

and \[F(x) = f(x) - f^{(2)}(x) + f^{(4)}(x) - …​ + (-1)^n f^{(2n)}(x)\] for every positive integer \(n\).

First note that \(f(x)\) and its derivatives \(f^{(i)}(x)\) have integral values for \(x = 0\), and also for \(x = \pi = a/b\) since \(f(x) = f(a/b - x)\).

We have \[\frac{d}{d{x}}(F'(x)\sin x - F(x) \cos x) = F''(x)\sin x + F(x) \sin x = f(x) \sin x\] whence \[ \int^{\pi}_{0} f(x) \sin x dx = [F'(x) \sin x - F(x) \cos x]^\pi_0 = F(\pi) + F(0) \in \mathbb{Z} \] But for \(0 < x < \pi\), we have \[ 0 < f(x) \sin x < \frac{\pi^n a^n}{n!} \] which means we have an integer that is positive but tends to zero as \(n\) approaches infinity, which is a contradiction.

Proof due to Ivan Niven.