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I'm puzzling a bit over an example I read in Rudin'sPrinciples of Mathematical Analysis. He has just defined least upper bound in the section I am reading, and now he wants to give an example of what he means. So the argument goes like this:

Consider the set A, where A = {p} s.t. p^{2}< 2 and p [itex]\in Q+[/itex] the set B, where B = {p} s.t. p^{2}> 2 and p is the same as above.

Now let A [itex]\subset Q[/itex] and B [itex]\subset Q[/itex], where Q is the ordered set of all rational numbers. He says that A has no least upper bound and B has no greatest lower bound.

I do not see why.

If I consider A by itself a subset of Q, then I think 2 = sup A, and B by itself a sub set of Q, 2 = inf B.

I could see that if we are talking about the set A AND B, then there is no sup A, if A [itex]\subset A AND B[/itex], because he just proved that there is no least element of B and no greatest element of A, and so it follows there is could be neither sup A nor inf B inthiscase.

But he states to consider A and B as subsets of Q.

Any help clarifying this matter would be greatly appreciated.

Also, sorry if this is in the wrong place; not sure where it goes, so I figured general math would be best.

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# Bounds on Sets

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