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Thread: more help upper bounds

  1. #1
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    more help upper bounds

    1) Let A and B be a subsets of the real numbers with least upper bound u and v. Prove that their union has a least upper bound, and express it in terms of u and v.

    2) Let A be the set of negative real numbers. Prove that 0 is equal to the least upper bound of A.

    Hint: one needs to check that 0 is an upper bound and if x < 0 then 0 is not an upper bound; i.e., there is some y in A such that x < y.


    I am soo sorry to keep on asking questions, but Math is reallly hard for me. Thank you very much in advance.
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  2. #2
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    Quote Originally Posted by jenjen View Post
    2) Let A be the set of negative real numbers. Prove that 0 is equal to the least upper bound of A.
    Assume, $\displaystyle A$ has a least upper bound $\displaystyle x$.

    By trichtonomy,
    $\displaystyle x<0$
    $\displaystyle x=0$
    $\displaystyle x>0$

    If, $\displaystyle x<0$
    Then, $\displaystyle x/2<0$ element of $\displaystyle A$
    And, $\displaystyle x/2>x$
    A contradiction.

    If, $\displaystyle x>0$
    Then, $\displaystyle x/2>0$ is an upper bound.
    And, $\displaystyle x/2<x$.
    A constradiction.

    Thus, $\displaystyle x=0$ if the least upper bound exists.
    Which is true because any negative is strictly less than a positive.

    1) Let A and B be a subsets of the real numbers with least upper bound u and v. Prove that their union has a least upper bound, and express it in terms of u and v.
    Use the completeness property. $\displaystyle A,B$ are upper bounded. Then, $\displaystyle A\cup B$ is upper bounded. Thus, by completeness it has a least upper bound.
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  3. #3
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    thanks ThePerfect Hacker but I have a question.

    After we use the completeness property, then I we concluded that A U B has a least upper bouard but how do I express it in terms of u and v like how the question was worded?
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