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Homework Help: Bounded sequence

  1. Feb 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Let [tex]x_m = 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{m}, m \in N[/tex].
    Prove [tex]x_m[/tex] is not bounded above and therefore [tex]x_m[/tex] does not converge.

    2. Relevant equations
    We know from our class an important theorem stating that:
    If sequence converges then the sequence is bounded.

    Thus we can say if the sequence is not bounded then it is not convergent.

    3. The attempt at a solution
    By above (#2), i just have to show our sequence is not bounded. This means i have the following claim:
    [tex]x_m[/tex] is not bounded above if and only if given any S > 0 , there exists m such that [tex]x_m[/tex] > S.

    1. Do i have to prove both sides of the arguement (if and only if)? Or can I just change my claim to a one sided (left to right)?

    2. Can someone help me formulate some thoughts on how to begin this proof?
  2. jcsd
  3. Feb 21, 2010 #2


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    Homework Helper

    This is a lot like your other post. Here's a hint. 1/3+1/4>1/2. 1/5+1/6+1/7+1/8>1/2. 1/9+1/10+...+1/15+1/16>1/2. Why?
    Last edited: Feb 21, 2010
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