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Math Help - Upper Bounds and Supremums

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    Upper Bounds and Supremums

    Suppose that A is contained in the set of all real numbers and is bounded above. Prove that if A contains one of its upper bounds, then this upper bound is sup A.

    It seems like it should be a simple problem, but I'm lost.
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    Re: Upper Bounds and Supremums

    Quote Originally Posted by lovesmath View Post
    Suppose that A is contained in the set of all real numbers and is bounded above. Prove that if A contains one of its upper bounds, then this upper bound is sup A.
    Suppose that a \in A\; \wedge \;\left( {\forall x} \right)\left[ {x \in A \Rightarrow x \leqslant a} \right].

    Now prove that a=\sup(A).

    Hint: If you assume that a<\sup(A) there is an intermediate contradiction.
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