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Boundedness homework help

  1. Oct 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [itex]\ell_\infty \mathbb({R})[/itex] be the set of bounded real sequences with k > 0 such that [itex]\left | x_n \right |\le k[/itex]

    a) [itex](n)=(1,2,3...) \notin \ell_\infty \mathbb({R})[/itex]. This is not bounded 'above'?

    b) [itex](2n^2+1) \notin \ell_\infty \mathbb({R})[/itex] Same answer as above?

    c) [itex](1/n)=(1,1/2,1/3,1/4...) \in \ell_\infty \mathbb({R})[/itex] Is bounded above?

    d) [itex](4-1/n) \notin \ell_\infty \mathbb({R})[/itex] Why is this not bounded? Is it because the value wll not go below 0?
  2. jcsd
  3. Oct 31, 2011 #2


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    Science Advisor

    Re: Boundedness

    a) yes.

    b) yes.

    c) bounded above AND below: 0 < 1/n ≤ 1

    d) i think there's a typo here
  4. Oct 31, 2011 #3


    Staff: Mentor

    Re: Boundedness

    Correct. No matter how large an M you pick, for some n, an > M.
    Yes, by 1.
    Looks bounded to me. Every number in the sequence is less than 4. Why do you think it's not bounded?
  5. Nov 1, 2011 #4
    Re: Boundedness

    Thanks guys,

    Is d) bounded above AND below....because [itex]0<(4-1/n) \le 4[/itex]...?
  6. Nov 1, 2011 #5


    Staff: Mentor

    Re: Boundedness

    For d, you have 3 <= 4 - 1/n < 4, with n being a positive integer.
  7. Nov 1, 2011 #6
    Re: Boundedness

    Thanks Mark.
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