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Math Help - Supremum and infimum

  1. #1
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    Supremum and infimum

    Find the supremum and infimum of the following sets:

    1) \left\{\frac{n}{1+n^2} : n=1,2...\right\}

    For this would you just start to sub numbers in:

    \{\frac{1}{2},\frac{2}{5}\}

    And look at the limit as n approaches infinity

    \displaystyle\lim_{n\to\infty}\frac{n}{1+n^2}=0

    So therefore the supremum is \frac{1}{2} and the infimum is 0

    2) \left\{\frac{n}{1+n^2} : n\in\mathbb{Z}\right\}

    Once again would you just substitute numbers to see what is happening so the supremum and infimum would be:

    -\frac{1}{2} and \frac{1}{2} respectively


    I'm having some trouble with this one:


    3) \{x\in\mathbb{Q} : x^2<2\}
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Find the supremum and infimum of the following sets:
    I'm having some trouble with this one:
    3) \{x\in\mathbb{Q} : x^2<2\}
    Consider \pm\sqrt2
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  3. #3
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    \sqrt{2} and -\sqrt{2} are not in \mathbb{Q}, of course, but the supremum and infimum of a set don't have to be in the set.
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