Seems begging for contradiction.

Suppose two are rational. Let's take and .

Those can be expressed as ratios of integers.

and

Solving simultaneously shows e is rational AND is rational, since:

1) Looking at denominators, they must be divisible by 2bd and

2) Looking at numerators, integers are closed over addition, subtraction, and multiplication.

I certainly have not covered the whole proof, but that seems a good piece of it.