E Is Irrational 

$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots$
So, suppose that $e=m/n.$
Multiply both sides by $n!$.
The lefthand side becomes an integer.
The first $n+1$ terms of the righthand side become integers.
The rest of the righthand side is $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots$ which is positive but smaller than 1 and therefore not an integer.
So, integer = integer + notinteger; contradiction.
Enrichment task
Prove that $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots<1$