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Math Help - Supremum Question

  1. #1
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    Supremum Question

    Suppose A and B are non-empty sets that are both bounded above, and that for all X in A, there exists a Y in B such that x is less than or equal or y.

    a) Prove that sup A is less than or equal to sup B (Need a general proof)
    b) Do A = (6/7, 1) and B = {n/(n+1): n in N} satisfy the conditions? Explain.
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  2. #2
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    Quote Originally Posted by Janu42 View Post
    Suppose A and B are non-empty sets that are both bounded above, and that for all X in A, there exists a Y in B such that x is less than or equal or y.
    a) Prove that sup A is less than or equal to sup B (Need a general proof)
    b) Do A = (6/7, 1) and B = {n/(n+1): n in N} satisfy the conditions? Explain.
    We know that \alpha  = \sup (A)\,\& \,\beta  = \sup (B) both of these exist.
    Now suppose that \beta < \alpha. That means that \beta is not an upper bound for A.
    That means  \left( {\exists x \in A} \right)\left[ {\beta  < x \leqslant \alpha } \right]
    But by the given  \left( {\exists y\in B} \right)\left[ {x \leqslant y} \right]
    Do you see the contradiction?
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  3. #3
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    sup(B) has to be an upper bound for A, correct? That's the contradiction? Because by the given information we cannot have the case where sup(B) is not an upper bound for A. Therefore, sup(A) has to be less than or equal to sup(B).
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