Help with some proofs involving bounds.

[daon]

The "Archimedean Property." I need to prove that the set of Naturals, considered as a subset of the Reals, is not (fixed) bounded above.

So I know the definitions and everything but I can't make any useful connections. I know this will most likely be a proof by contradiction involving a LUB. So..

Assume BWOC that the set of Natural numbers is Bounded above. If it is bounded above, then there is a LUB. Let x = LUB(N). From this I know that for all y \(\displaystyle \in\) N, y <= x;

...Here is where I get stuck. I think it may be helpful to prove that there must exist a natural number between x and x+1, thus contradicting that x is a LUB, but am not sure how to do that, or even if that is the right path to take.

Thanks in advance for any help.

 

[pka]

The "Archimedean Property." I need to prove that the set of Naturals, considered as a subset of the Reals, is bounded above.

FIRST. That is NOT the "Archimedean Property." It just is not.
The Archimedean Property states that: If each a and b is positive real number then the is a positive integer N such that aN>b.

It is impossible to prove: “the set of Naturals, considered as a subset of the Reals is bounded above.” Because that is false.

 

[daon]

The "Archimedean Property." I need to prove that the set of Naturals, considered as a subset of the Reals, is bounded above.

FIRST. That is NOT the "Archimedean Property." It just is not.
The Archimedean Property states that: If each a and b is positive real number then the is a positive integer N such that aN>b.

It is impossible to prove: “the set of Naturals, considered as a subset of the Reals is bounded above.” Because that is false.

I meant is not bounded above. Sorry! Also, the book states the Archimedean Property as this along with two Corollaries which I have to prove as well:

1) Given a real M there is an a Natural number n such that n > M.
2) Given a real number epsilon, there is a Natural number n such that 1/n < epsilon.

But I am mostly concerned with the first part as I can't try these corollaries unless I prove that N is not bounded above, right?

 

[pka]

Now I am really confused!
Is that really what your text says is the statement of the "Archimedean Property"?
If it is, how odd?

Here is a standard proof.
Lemma: If k is a integer the k<k+1; proof: 0<1 add k then k<k+1.

So suppose that α is the LUB of the set N.
Then α-1 is not an upper bound of N. That means that there is some j in N such α-1<j≤α. Add 1 and get α<j+1. But j+1 is in N and it is greater than the LUB of N. That is a contradiction.