Thread: Supremum

Suppose S (a subset of R) is an upper bounded set and u is in S. Assume u is an upper bound for S. Show that supS=u.

This seems to be just a definition so how would you prove it? Also, doesn't u have to be the least upper bound (and not just some upper bound)?

Originally Posted by JaysFan31

Suppose S (a subset of R) is an upper bounded set and u is in S. Assume u is an upper bound for S. Show that supS=u.

This seems to be just a definition so how would you prove it? Also, doesn't u have to be the least upper bound (and not just some upper bound)?

Suppose u is not the supremum, then there is another v which is greater (or
equal) every element of S, but less than u. But u is in S and so we have the
contradiction:

u<=v<u.

So u is the supremum.

RonL

Originally Posted by CaptainBlack

Suppose u is not the supremum, then there is another v which is greater (or
equal) every element of S, but less than u. But u is in S and so we have the
contradiction:

Am I missing something? v is not necessarily an element of S. Therefore saying that u>v (because it is an upper bound) does not apply

Your proof falls as dramatically as the Roman Empire.

Originally Posted by ThePerfectHacker

Am I missing something? v is not necessarily an element of S. Therefore saying that u>v (because it is an upper bound) does not apply

u>v, because if u is not the supermum there is an upper bound
which is less than u, and by supposition one such is v.

Your proof falls as dramatically as the Roman Empire.

Since the Roman Empire took from sometime in the 5th or 6th century
up to 1453 and the death of Constantine XI, other than the last moments
long drawn out is a more appropriate description of the death of the
Emprire.

RonL

Originally Posted by CaptainBlack

Since the Roman Empire took from sometime in the 5th or 6th century
up to 1453 and the death of Constantine XI, other than the last moments
long drawn out is a more appropriate description of the death of the
Emprire.

RonL

Touche! Maybe as fast as the USSR would be a better description?

Originally Posted by Jameson

Maybe as fast as the USSR would be a better description?

I do not like that simile.
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I would have said as "...Greece" but I like Greece thus I chose not to choose it.