## Proof that Pi is Irrational

Suppose $\pi = a/b$. Define

and

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

whence

But for $0 \lt x \lt \pi$, we have

which means we have an integer that is positive but tends to zero as $n$ approaches infinity, which is a contradiction.