To prove that e is irrational
Now each term of an is positive, so an > 0.
The terms of bn are a geometric series, so
Moreover in comparing each term of an with the corresponding term of bn we have
etc, so it follows that an < bn.
Hence we have .
Now suppose e is rational, ie that
Then we have
We can choose n so that q divides n (eg n = q).
Then the expression enclosed by inequality signs is an integer.
An integer cannot lie between 0 and .
That is a contradiction, so the assumption (that e is rational) can only be false.
Hence finally e is irrational.