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Least Upper Bound proof

  1. Oct 19, 2009 #1
    Least Upper Bound proof....

    1. The problem statement, all variables and given/known data
    Suppose A is a nonempty set that has x as an upper bound. Prove that x is the least upper bound of the set A iff for any E>0 there exists a y in A such that y>x-E


    2. Relevant equations
    None


    3. The attempt at a solution
    The forward where you assume that x is the least upper bound is very easy, but I'm having some trouble proving the reverse.....

    This is what I have so far....

    Let x be an upper bound of A, and choose a point z in A.
    If x is an upper bound of A, then x+z is also an upper bound.
     
  2. jcsd
  3. Oct 19, 2009 #2

    LCKurtz

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    Re: Least Upper Bound proof....

    To prove the reverse, you are given that x is an upper bound for A having the property:

    If [itex]\epsilon > 0[/itex] there is a y in A satisfying x - [itex]\epsilon[/itex] < y

    You have to show that no number z < y is an upper bound for A. What problem would arise if there was such a number z?

    [Edit] Sorry, there is a typo. The last paragraph should have read:

    You have to show that no number z < x is an upper bound for A. What problem would arise if there was such a number z?
     
    Last edited: Oct 20, 2009
  4. Oct 20, 2009 #3
    Re: Least Upper Bound proof....

    Hmm.... if there were such a number z, then y could not be the least upper bound.....

    Could the proof go something like this?:

    Choose arbitrary E>0, and let y be an upper bound of A

    Suppose z is an upper bound of A, and y>z>y-E.

    y is not the lub

    does this finish the proof?
     
  5. Oct 20, 2009 #4

    LCKurtz

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    Re: Least Upper Bound proof....

    No. Sorry, but I had a typo which I have corrected. Read my reply and try again.
     
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