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Real Analysis Supremum of a Set Proof

  1. Feb 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Let S be a non empty set that is bounded about and β = sup S. Prove that for each ε > 0 there exists a point x in S such that x > β - ε.


    2. Relevant equations



    3. The attempt at a solution

    I don't really know how to begin this. I know it's true; I'm looking at the problem and I'm like, "Well, duh," but I can't prove it. I know that x ≤ β, and that β - ε ≤ x. Is there more that I have to do?


    edit: never mind. I'm an idiot. Proof by contradiction makes this really easy.
     
    Last edited: Feb 12, 2012
  2. jcsd
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