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:__

http://i40.tinypic.com/v8o2e0.jpg

Re: One Question about Proof on Irrationality of e

Quote:

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:__ http://i40.tinypic.com/v8o2e0.jpg

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.