To prove that e is irrational

 

We define

 

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 .

 

By definition,

.

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.

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