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Math Help - Completeness Axiom

  1. #1
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    Completeness Axiom

    A= \left\{ {x\in \mathbb{R}:x^3 < 2} \right\}

    Prove carefully that A has a least upper bound.

    A is non-empty and bounded above therefore by the completeness axiom the supremum of A exists. Can you get me started please?

    Also is maximum and least upper bound of a set the same thing?

    Thanks
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  2. #2
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    Supremum and LUB are two words for the same thing. LUB and maximum are not the same concept, since a maximum of a set must reside inside the set, whereas the LUB is not required to.

    My guess is that what they are looking for is a proof that A is non-empty and bounded above, so that the LUB exists. Probably the main point is showing that it is bounded above.
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  3. #3
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    Quote Originally Posted by charikaar View Post
    A= \left\{ {x\in \mathbb{R}:x^3 < 2} \right\}

    Prove carefully that A has a least upper bound.

    A is non-empty and bounded above therefore by the completeness axiom the supremum of A exists. Can you get me started please?

    Also is maximum and least upper bound of a set the same thing?

    Thanks
    As already said, you must first prove A is non-empty and bounded above, which seems to be fairly simple. Now take a look at w:=\sqrt[3]{2} and prove this number is the LUB:

    1) First, prove w is an upper bound;

    2) Prove that \forall \epsilon>0\,\,\exists\,a_{\epsilon}\in A\,\,s.t.\,\,w-\epsilon<a_{\epsilon}\leq w

    Tonio
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