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Intro to Analysis (Boundedness of Cauchy)

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove the Boundedness Theorem for Cauchy Sequences by a contrapositive argument.

    2. Relevant equations



    3. The attempt at a solution

    If {an} is not bounded then {an} is not a cauchy sequence.

    We will prove that {an} is not a cauchy sequence by showing that there exists an ε>0 so that for every pos. int. N there exists an n,m>N so that abs(an-am)≥ε.

    Set ε=1
    Consider positive N arbitrary.
    Since by hypothesis {an} is not bounded to mean for every B there exists an n so that abs(an)>B. Choose such an n.

    This is where I am stuck, how do I find/choose an m? Also how do I get the inequalities to work out at the end?
     
  2. jcsd
  3. Mar 6, 2012 #2
    The same way you found n. You know that {a_n} is unbounded, so for any B, there exists something greater than it, right? well, what about a_n? Shouldn't there be something greater than a_n?

    Also, look at some consequences of the triangle inequality.
     
  4. Mar 6, 2012 #3
    Alright I am following what you are saying but get stuck at:

    abs(an-am) ≥ abs(an) - abs(am)

    Then I do not know how to make it bigger than one.
     
  5. Mar 6, 2012 #4
    So take a peek at the reverse triangle inequality. There's another one, like you have, but a little different.

    But if we know that we can some large a_n, well, since {a_n} is unbounded, there exists some |a_m| > |a_n|, right? Then certainly there's some |a_m| > |a_n| + 1?
     
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