Pi is Irrational/Proof 2

Theorem
Pi ($\pi$) is irrational.

Proof
For any fixed number $q$, let:


 * $\displaystyle A_n = \frac {q^n} {n!} \int_0^\pi \left[{x\left({\pi - x}\right)}\right]^n \sin x \ \mathrm d x $

Integration by Parts twice gives:

Writing $\left({\pi - 2x}\right)^2 = \pi^2 - 4x \left({\pi - x}\right)$ gives the result:


 * $(1): \quad \displaystyle A_n = \left({4n - 2}\right) q A_{n - 1} - \left({q\pi}\right)^2 A_{n - 2}$

Suppose $\pi$ is rational. Then $\pi = \dfrac p q$ where $p$ and $q$ are integers and $q \ne 0$. We will deduce that $A_n$ is an integer for all $n$.

First confirm by direct integration that $A_0$ and $A_1$ are integers:


 * $\displaystyle A_0 = \int_0^\pi \sin x \ \mathrm d x = 2$


 * $\displaystyle A_1 = q \int_0^\pi x\left({\pi - x}\right) \sin x \ \mathrm d x = 4q$

Suppose $A_{k - 2}$ and $A_{k - 1}$ are integers. Then by $(1)$ and the assumption that $q$ and $q\pi$ are integers, $A_k$ is also an integer. So $A_n$ is an integer for all $n$ by Second Principle of Mathematical Induction.

For $x \in \left[{0, \pi}\right]$, we have $0 \le \sin x \le 1$, and $0 \le x\left({\pi - x}\right) \le \pi^2/4$, hence:


 * $\displaystyle 0 < A_n < \pi \frac {\left(q\pi^2/4\right)^n} {n!}$

From Power over Factorial:


 * $\displaystyle \lim_{n \to \infty} \frac {\left(q\pi^2/4\right)^n} {n!} = 0$

It follows from Squeeze Theorem that:


 * $\displaystyle \lim_{n \to \infty} A_n = 0$.

Hence for sufficiently large $n$, $A_n$ is strictly between $0$ and $1$.

This contradicts that $A_n$ is an integer.

It follows that $\pi$ is irrational.