Thread: One Question about Proof on Irrationality of e

Dear all,

I'm looking at this particular proof that is irrational. Please find it beneath my writing.

This proof by contradiction assumes that . I understand this.

However, while proving that is the contradiction, the proof uses the second boxed inequality
where



and

and so on...

However, although , it is entirely possible for AND since .

In this case then the , so the above steps and the proof itself fail?

Thank you very much for your assistance.

Proof:

Originally Posted by scherz0

Dear all,

I'm looking at this particular proof that is irrational. Please find it beneath my writing.

This proof by contradiction assumes that . I understand this.

However, while proving that is the contradiction, the proof uses the second boxed inequality
where



and

and so on...

However, although , it is entirely possible for AND since .

In this case then the , so the above steps and the proof itself fail?

Thank you very much for your assistance.

Proof:

It should be pretty obvious that q is a positive integer, because of how q is defined in a and b.

And since the proof involves showing that e = p/q, and it can easily be seen that e > 0, since q > 0, then so must be p.