e is irrational

#Problem 8

Prove that e is irrational

Solution scheme and approach
e=1+1/1!+2/2!+3/3!+.....(i)
Let e is rational so e=p/q
multiplying equation number (i) by q! we get
q!*e=q!+q!/1!+q!/2!+q!/3!+....+q/q!+ other terms
Now as q!+q!/1!+...+q!/q! is an integer and q!*e is also an integer
So we can say other terms should be an integer. Let I signifies the rest terms whose summation is an integer.
I=q![1/(q+1)!+1/(q+2)!+1/(q+3)!+...]
=1/(q+1)+1/(q+1)*(q+2)+1/(q+1)*(q+2)*(q+3)+.....
<1/(q+1)+1/(q+1)^2+1/(q+1)^3+....=1/(q+1)/[1-1/(q+1)]=1/q<1
=>I< 1
Hence contradiction.
So, e should be an irrational number. Q.E.D.
[This solution has been proposed by Fourier]