Prove that there is no positive integer between 0 and 1

Theorem : Prove that there is no positive integer between 0 and 1

<proof>

Suppose there is a positive integer 'a' between 0 and 1. Let S = {n E Z+ | o < n <1}. Since 0 < a < 1, a E S, so S in nonempty.

</proof>

<me> So far fine from side</me>

<proof>

Therefore, by well ordering principle, S has a least element 'l', where o < l < 1

</proof>

<me> fine from my side</me>

<proof>

Then 0 < l^2 < l, so l^2 E S.

</proof>

<me>

This is not fine. First of all from the set S is supposed to hold only +ve integers, given by the author as

S = {n E Z+ | 0 < n 1}. And the square of a no is less than the original no, only if the no is less than 1. But when the author has assume that 'S' contains only +ve integers Z+, he can't assume that 'l' is a non-integer which he has done in this case by saying 0 < l^2 < l.

Here is the rest of the proof anyways. If I am wrong I would greatly appreciate your help in understanding why I am wrong.

</me>

<proof>

But l^2 < l, which contradicts our assumption that 'l' is a least element of S.Thus there are +ve integers

</proof>