Proving a number is irrational is mostly done by contradiction. So first suppose e is rational: e = p/q where p, q are coprime integers.
We know that q≥2 as e is not an integer (in fact, it’s in between 2 and 3). Then
Note that, as q!e and n are natural numbers, we must have that x is a natural number.
And so we can bound x in the following way
This is a contradiction since q!e must be a natural number, but it is a sum of an integer n plus a non-integer x. Hence, e is irrational.