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Math Help - Prove that s is the sup of A.

  1. #1
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    Prove that s is the sup of A.

    Could some one point me in the right direction with this proof to get me started? I'm not sure where to begin.

    If a set A ⊆ R contains one of its upper bounds s, prove that s is the supremum of A.
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  2. #2
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    Write out explicitly:

    1. The definition of an upper bound.
    2. The definition of a supremum.

    It will help you think this through more clearly.
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  3. #3
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    Quote Originally Posted by Mazerakham View Post
    Write out explicitly:

    1. The definition of an upper bound.
    2. The definition of a supremum.

    It will help you think this through more clearly.
    okay. So taking your advice I have:

    An upper bound: If a set A⊆R a real number s is an upper bound for A if for all x \inA, x \leqs.


    a supremum: a real number s is least upper bound for A if s is an upper bound for A having the property that, if b is also an upper bound for A then s \leqb.

    So am I showing that there is a y such that y>s- \epsilon or is that still just the definition. I'm sorry I think it is the wording of the question that is throwing me off.
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  4. #4
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    Is this a step in the right direction?

    If t is another upperbound for A, then t \geq x for any x \in A. Since s \in A, we must have that t \geq s. So therefore s is the smallest upper bound or least upper bound.
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  5. #5
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    Quote Originally Posted by alice8675309 View Post
    Is this a step in the right direction?

    If t is another upperbound for A, then t \geq x for any x \in A. Since s \in A, we must have that t \geq s. So therefore s is the smallest upper bound or least upper bound.

    Right on the nose, and thus s is in fact the maximum of A as A contains it.

    Tonio
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  6. #6
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    Yep. I sketch could have helped too. Never try to do a symbolic proof without a picture in your head! The statement is of course intuitively true if you visualize it, and the proof should be connected to that intuition. Nice job.
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