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Math Help - supremum

  1. #1
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    supremum

    Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by inthequestofproofs View Post
    Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?
    I assume, as your last posts indicated, that you are dealing with closed subsets of \mathbb{R}. Then the answer is yes. Notice that \overline{E}=\left\{x\in\mathbb{R}:d(x,E)=0\right\  }. But, if E is closed then \overline{E}=E. So, let me ask you: what is d\left(E,\sup\text{ }E\right)?

    (if the supremum exists)
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  3. #3
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    answer to your question

    It would be zero, and thus, it is in the closed set
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by inthequestofproofs View Post
    It would be zero, and thus, it is in the closed set



    Now prove it
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  5. #5
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    I wanted to ask what about the set

    M=[0,\inf[, which is a closed subset of \mathbb{R}.

    Then does the supremum of M exist?

    if so then sup\{M\}=\inf but would you say that it is in M ?

    What about the empty set? it is both open and closed ?

    what is sup\{\varnothing\} ? - \inf ?
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  6. #6
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    In order to have a supremum, a set must first have an upper bound. [0, \infty[ does not have an upper bound so no supremum.
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