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Prove alpha=sup(S) is equivalent to alpha belongs to S closure

  1. Given that alpha is an upper bound of a given set S of real numbers, prove that the following two conditions are equivalent:
    a) We have alpha=sup(S)
    b) We have alpha belongs to S closure

    I'm trying to prove this using two steps.
    Step one being: assume a is true, then prove b is true.
    Step two being: assume b is true, then prove a is true.

    Could anyone help me with step two?
    Assuming alpha belongs to S closure.....
  2. jcsd
  3. radou

    radou 3,217
    Homework Helper

    If I remember right, I think I gave you a useful condition for a point to be in the closure of a set. Do you see how you can use it here?
  4. radou

    radou 3,217
    Homework Helper

    Of course, your "steps" are a correct way to prove equivalence of statements, from a logical point of view.
  5. No I don't see how I can use it here in this problem.

    How would I start my step two? I know I assume alpha belongs to S closure, but I am not sure where to go from there.
  6. radou

    radou 3,217
    Homework Helper

    A point x is in the closure of a set A if any neighbourhood of x intersects A. Now, what is an important property of the supremum (involving "ε")?
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