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Math Help - Help finding suprema and infima

  1. #1
    Senior Member Danneedshelp's Avatar
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    Help finding suprema and infima

    Q: Compute, without proofs, the suprema and infima of the following sets:

    (a) \{n\in\mathbb{N}:n^{2}<10\}

    (b) \{\frac{n}{n+m}:m,n\in\mathbb{N}\}

    (c) \{\frac{n}{2n+1}:n\in\mathbb{N}\}

    (d) \{\frac{n}{m}:m,n\in\mathbb{N} with m+n\leq\\10\}

    A:

    (a) \{n\in\mathbb{N}:n^{2}<10\}=[0,3]. So, the Sup(A)=3 and Inf(A)=0

    (b) I think the interval is (0,1), but I am not sure.

    (c) \{\frac{n}{2n+1}:n\in\mathbb{N}\}=[\frac{1}{3},\frac{1}{2}). So, Sup(A)=1/2 and Inf(A)=1/3

    (d) \{\frac{n}{m}:m,n\in\mathbb{N} with m+n\leq\\10\}. So, Sup(A)=9 and Inf(A)=1/9

    I am confused by (b) mostly, but I am not sure about the rest of them either. Do the sup and inf have to be natrual numbers?
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by Danneedshelp View Post
    Do the sup and inf have to be natrual numbers?
    In this context I don't believe they do - but it has not been specified in what set they are allowed to be. In this context it ought to be \mathbb Q (the rationals).

    So I would say that for b the sup is 1 and inf is 0 as you suggest.

    For the sup, set m=0 and let n be anything you like, you've got 1 and nothing you can do can make it bigger.

    Second, set n = 0 and let m = anything you like, you've got 0.

    Are you using the convention that \mathbb N = \{1, 2, 3, ...\} or \mathbb N = \{0, 1, 2, ...\}? In the first case the answers will be the same but you won't have 0 and 1 actually in the sets described.
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Matt Westwood View Post
    In this context I don't believe they do - but it has not been specified in what set they are allowed to be. In this context it ought to be \mathbb Q (the rationals).

    So I would say that for b the sup is 1 and inf is 0 as you suggest.

    For the sup, set m=0 and let n be anything you like, you've got 1 and nothing you can do can make it bigger.

    Second, set n = 0 and let m = anything you like, you've got 0.

    Are you using the convention that \mathbb N = \{1, 2, 3, ...\} or \mathbb N = \{0, 1, 2, ...\}? In the first case the answers will be the same but you won't have 0 and 1 actually in the sets described.
    \mathbb N = \{1, 2, 3, ...\}

    My bad.

    So the first one goes from 1 to 3?

    How do the others look?

    Thanks a lot.
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  4. #4
    Super Member Matt Westwood's Avatar
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    They look fine to me. You're all right, you know what you're doing here.
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