suprema and infima of sets without proof - Abbott 1.3.8

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Your reasoning for (a) is incorrect: if you fix $m=1$ and let $n \rightarrow \infty,$ we get $1/n \rightarrow 0.$ So $\inf A = 0.$

The others seem fine. Usually in analysis $\mathbb N$ does not include $0$ and judging by (b) I would assume this convention is taken here. It is likely that the author lists his notational conventions near the start of the book, so you should check there.

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Dando18
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Updated on August 01, 2022

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  • Dando18
    Dando18 over 1 year

    This problem comes from Understanding Analysis by Abbott.

    1.3.8 Compute, without proofs, the suprema and infima (if they exist) of the following sets:

    (a) $ \{ m/n : m,n \in \mathbb{N} \text{ with } m<n \} $

    (b) $ \{ (-1)^m / n : m,n \in \mathbb{N} \} $

    (c) $ \{ n/(3n+1) : n \in \mathbb{N} \} $

    (d) $ \{ m/(m+n) : m,n \in \mathbb{N} \} $

    Assume $A$ represents the set in context. My answers:

    (a) I believe this should be $\sup A = 1$, because $m \ngeq n$ then $m/n$ can't be $1$ or greater, but it continuously gets closer. I wasn't sure here either $\inf A = 0$ or $\inf A = 1/2$ I think depending on the inclusion of $0$.

    (b) This has $\min \{A\} = -1$ and $\max \{A\} = 1$, so $\inf A = -1$ and $\sup A = 1$.

    (c) The values seem to be approaching $1/3$, so I answered $\sup A= 1/3$, and again I'm not sure if $\inf A = 0$ or $\inf A = 1/4$.

    (d) $\inf A = 0$ (think huge $n$ and small $m$) and $\sup A = 1/2$ or $\sup A = 1$ depending on whether $n$ can be $0$ or not.

    Any corrections?

    • Mitchell Faas
      Mitchell Faas over 6 years
      As far as I'm concerned, 0 is a natural number. How you feel about it is up to you though, but generally everything is more readable if you pick one and stick with it (for a certain solution). But at (d), if we assume $n>0$, then $\sup A =1$ for what if we set $n=1$ and $m\rightarrow\infty$.
    • Nate Eldredge
      Nate Eldredge over 6 years
      For Abbott, 0 is not in the set $\mathbb{N}$. It's defined on page 2.
    • abhishek
      abhishek over 2 years
      (a) $\sup A = 1$ and $\inf A = 0$
    • abhishek
      abhishek over 2 years
      (b) $\sup A = 1$ and $\inf A = -1$
    • abhishek
      abhishek over 2 years
      (c) $\sup A = \dfrac{1}{3}$ and $\inf A = \dfrac{1}{4}$
    • abhishek
      abhishek over 2 years
      (d) $\sup A = 1$ and $\inf A = 0$
  • Dando18
    Dando18 over 6 years
    thanks for the answer and tip! I checked and the book does not include $0$.