Presume for contradiction that e is rational, therefore
Such that p and q are integers. Since q is an integer there must be the term in the series of e, thus
Multiply both sides by q to get
The term p(q-1)! must be an integer as is since q! is divisible by all factorials including q!.
It follows that
The left side is greater than zero. The right side
is an infinite series with
demonstrating that is not an integer. Thus
for integer p and q. Therefore e is irrational. Q.E.D.