>Prove that e-(1+1/1!+1/2!+1/3!+.......+1/n!)<1/(n!n) for all
>natural number n.
>
>Hence prove that e is an irrational number !
>

Assume e is rational M/N in lower terms. Then

M/N - (1+1/1!+1/2!+1/3!+.......+1/N!)<1/(N!N), or

M·N!/N - (N!+N!/1!+N!/2!+N!/3!+.......+N!/N!)<1/N.

M·N!/N is a whole number, as is the the number in parentheses. Since their difference is an integer, the inequality implies that the two are equal:

M·N!/N = N!+N!/1!+N!/2!+N!/3!+.......+N!/N!, or

e = 1+1/1!+1/2!+1/3!+.......+1/N!,

which contradicts the fact that

e = 1+1/1!+1/2!+1/3!+.......+1/N! + a little something.

Therefore, our assumption that e = M/N, leads to a contradiction.