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Math Help - proving convergence

  1. #1
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    proving convergence

    Let
    A be a bounded subset of R with infinitely many elements. If

    s
    = supA is NOT an element of A, prove that there exists a strictly
    increasing sequence (
    xn), xn inA such that limit as n approaches infinity of (xn) goes to s.

    I know how to prove that the limit of a strictly increasing sequence goes to the supremum, but I'm not sure how to show that there must exist a strictly increasing sequence if the supremum is not an element of the subset. Can someone show me how to do this? I would really really appreciate it.


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  2. #2
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    Quote Originally Posted by CindyMichelle View Post
    Let
    A be a bounded subset of R with infinitely many elements. If

    s
    = supA is NOT an element of A, prove that there exists a strictly
    increasing sequence (
    xn), xn inA such that limit as n approaches infinity of (xn) goes to s.

    I know how to prove that the limit of a strictly increasing sequence goes to the supremum, but I'm not sure how to show that there must exist a strictly increasing sequence if the supremum is not an element of the subset. Can someone show me how to do this? I would really really appreciate it.


    (Note, the set must be infinite because the supremum of a finite set belongs to the set).

    Let \sup A = s. Choose \epsilon = 1 then s - 1 cannot be an upper bound so there exists s_1 \geq s - 1. Choose \epsilon = 1/2 then s-1/2 cannot be an upper bound so there exists s_2 \geq s - 1/2. Choose \epsilon = 1/3 then s-1/3 is not an upper bound so s_3 \geq s - 1/3. In generel s_k \geq s - \frac{1}{k} where s_k is this constructed sequence. Thus, |s_k - s| \leq \frac{1}{k}. Thus this sequence converges to s.
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