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Intro to Real Analysis: Supremum

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the supremum of E=(0,1)

    2. Relevant equations

    3. The attempt at a solution
    By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show [tex]1<=M[/tex]. Can I just say that any upper bound of M must be greater than or equal to one based on the definition of open interval again?

    I'm just not sure if that last line is completely correct. Thanks.
  2. jcsd
  3. Sep 18, 2011 #2
    Use your theorems.

    s is the least upperbound if for all [itex]\epsilon > 0[/itex], there exists an [itex]a \in (0, 1)[/itex] satisfying [itex]s - \epsilon < a[/itex].
  4. Sep 18, 2011 #3
    Since you are only required to find the sup(E) you can just state it.

    Hint: sup(E) ∉ E.
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