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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