Hi can I get some help with the following theorem I need to prove:

Let A be a subset of the ordered set X. If A has an upper bound then A has a least upper bound..

Thanks!

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- April 29th 2009, 10:42 PMchaoticmathleast upper bound property
Hi can I get some help with the following theorem I need to prove:

Let A be a subset of the ordered set X. If A has an upper bound then A has a least upper bound..

Thanks! - April 30th 2009, 04:14 AMShowcase_22Quote:

If A has an upper bound then A has a least upper bound.

From the question we know that A has an upper bound. Let's suppose A has two upper bounds and with .

( is also valid but i've decided to choose ).

Hence A has a least upper bound which is . - April 30th 2009, 04:26 AMPlato
What if , is the set of rationals.

Is bounded above by 2?

But can have a least upper bound? No!

So the statement if false without some more given conditions. - April 30th 2009, 04:28 AMShowcase_22
but isn't the least upper bound , it's just that isn't in the set?

- April 30th 2009, 04:34 AMPlato