Well if there were then since and , it follows that . Now you have that , so continue in this way to get a contradiction.
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.
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: