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Math Help - Supremum

  1. #1
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    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)?
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  2. #2
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    Quote Originally Posted by JaysFan31 View Post
    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
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    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.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    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
    Last edited by CaptainBlack; September 20th 2006 at 03:59 AM.
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    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?
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  6. #6
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    Quote Originally Posted by Jameson View Post
    Maybe as fast as the USSR would be a better description?
    I do not like that simile.
    ---
    I would have said as "...Greece" but I like Greece thus I chose not to choose it.
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