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Math Help - Proving equivalences

  1. #1
    Junior Member
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    Proving equivalences

    Let X\ne\varnothing and let c\in\mathbb R an upper bound for X. Prove that the following sentences are equivalent:

    a) c=\sup X.

    b) For all n>0 exists an element x\in X so that c-\dfrac1n<x\le c.

    First a) implies b): since c=\sup X, then the number c-\dfrac1n is not an upper bound of X then there exists x\in X so that c-\dfrac1n<x\le c. Is that enough?

    I'm having problems to prove b) implies a), how to proceed?
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  2. #2
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    Suppose c \neq \sup X. Then there exists an upper bound c' < c for X...
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