Use the WOP of N to prove that there is no such that 0<n<1

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- August 18th 2009, 12:50 PMflower3well ordering principle
Use the WOP of N to prove that there is no such that 0<n<1

- August 18th 2009, 01:59 PMputnam120
Well if there were then since and , it follows that . Now you have that , so continue in this way to get a contradiction.

- August 18th 2009, 02:37 PMPlato
I am a bit puzzled by the wording of this question.

I my experience the following is a standard theorem: .

The unique integer is of course the floor of .

I see what putnam120 is trying to do. But I an not at all sure where W.O. be used.

The theorem I quoted above with give the result at once.

PS. Of course, this is all mute if your course includes a rigorous construction of the natural numbers. - August 18th 2009, 03:18 PMputnam120
W.O. states that every set of non-negative integers has a least element. Thus since only contains non-negative integers it to must have a minimal element. So using the construction I started you show that the set doesn't have a minimal element.

I usually see defined as follows:

1)

2)

3)