Thanks to Alexander Farrugia at Quora for passing on this simple proof that e is irrational, that is understandable by a competent high schooler – apparently based on a version posted by Sridhar Ramesh several years ago.

The basic idea is to use the alternating series representation of $#1/e#$ to show that for all $#n#$, $#1/e#$ is between $#S_n=\frac{a}{n!}#$ and $#S_{n+1}=S_{n}\pm\frac{1}{(n+1)!}#$ and so between $#\frac{a}{n!}#$ and $#\frac{a\pm 1}{n!}#$ for some integer $#a#$. This of course means that there is no $#n#$ for which $#1/e#$ can be written as a fraction $#N/D#$ with denominator $#D=n!#$, but if $#1/e=N/D#$ then it can also be written as $#\frac{N(D-1)!}{D!}#$ and so can be written as a fraction with factorial denominator.

Source: (1002) A Simple Proof that e is Irrational – Farrugia Maths – Quora