Euler's Number is Irrational

Theorem:
Euler's number $$e\,\!$$ is irrational.

Proof by Contradiction:
Assume that $$e\,\!$$ is rational. Then, there exist coprime integers $$m$$ and $$n$$ such that: $$\frac{m}{n} = e = \sum_{i=0}^{\infty}\frac{1}{i!}$$

Multiplying both sides by $$n!\,\!$$, observe that: $$\frac{m}{n}n! = n!\sum_{i=0}^{\infty}\frac{1}{i!} = \left (\frac{n!}{0!} + \frac{n!}{1!} + \frac{n!}{2!} + \ldots + \frac{n!}{n!} \right ) + \left ( \frac{n!}{(n+1)!} + \frac{n!}{(n+2)!} + \frac{n!}{(n+3)!} + \ldots \right )$$

Observe that the quantity on the left must be an integer, as it is composed entirely of sums and differences of integral components. It must be positive, as it is equal to $$\frac{1}{(n+1)} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \ldots$$, which is strictly positive.

Thus, $$m(n-1)! - \left (\frac{n!}{0!} + \frac{n!}{1!} + \frac{n!}{2!} + \ldots + \frac{n!}{n!} \right )$$ must be a positive integer less than 1, a contradiction, so $$e\,\!$$ must be irrational.

Q.E.D.