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Thread: Supremum and infimum

  1. #1
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    Supremum and infimum

    Find the supremum and infimum of the following sets:

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

    For this would you just start to sub numbers in:

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

    And look at the limit as $\displaystyle n$ approaches infinity

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

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

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

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

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


    I'm having some trouble with this one:


    3) $\displaystyle \{x\in\mathbb{Q} : x^2<2\}$
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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) $\displaystyle \{x\in\mathbb{Q} : x^2<2\}$
    Consider $\displaystyle \pm\sqrt2$
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  3. #3
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    $\displaystyle \sqrt{2}$ and $\displaystyle -\sqrt{2}$ are not in $\displaystyle \mathbb{Q}$, of course, but the supremum and infimum of a set don't have to be in the set.
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