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Show sequence is bounded in an interval

  1. Sep 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that a sequence {a_n} is bounded if and only if there is an interval [c, d] such that {a_n} is a sequence in [c,d].

    2. Relevant equations

    A sequence {a_n} is bounded provided that there is a number M such that |a_n| <= M.

    3. The attempt at a solution

    http://img180.imageshack.us/img180/7856/001hep.th.jpg [Broken]

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    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 12, 2010 #2

    jgens

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    The first part looks fine. I'm having difficulty reading the second part, but it looks like you assume the c=-d, which isn't necessarily a valid assumption.
     
  4. Sep 12, 2010 #3
    For the second part I let c = -d because I was trying to end with a result such as the sequence being in the interval [-M, M]. If I leave it as the interval [c, d] then I'm not sure how to end with the conclusion that every element of the sequence is less than a number M.
     
  5. Sep 12, 2010 #4

    jgens

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    Consider max{|c|,|d|}.
     
  6. Sep 12, 2010 #5
    So I can define M to be max{ |c|, |d| }.

    Therefore, |a_n| <= M.

    Would this be all that I need to write for the proof in that direction?
     
  7. Sep 12, 2010 #6

    jgens

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    It might be nice if you write things out a little more explicitly, but yes, it's all that you need.
     
  8. Sep 12, 2010 #7
    I'm not really sure how I should be more explicit for the proof.
     
  9. Sep 12, 2010 #8

    jgens

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    [tex]-\max\{|c|,|d|\} \leq c \leq a_n \leq d \leq \max\{|c|,|d|\}[/tex]
     
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