- #1
res3210
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Hey guys,
I'm puzzling a bit over an example I read in Rudin's Principles 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. p2 < 2 and p [itex]\in Q+[/itex] the set B, where B = {p} s.t. p2 > 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 in this case.
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.
I'm puzzling a bit over an example I read in Rudin's Principles 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. p2 < 2 and p [itex]\in Q+[/itex] the set B, where B = {p} s.t. p2 > 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 in this case.
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.