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Math Help - Convergent Sequence

  1. #1
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    Convergent Sequence

    Hi, I think I am close to the answer for this but I am missing something:

    Suppose x_{n} is a sequence satisfying |x_{n+1}-x_{n}|<\frac{1}{2^n} for all n. Show that x_{n} is a convergent sequence.

    So to prove it is convergent, I need to prove it is a Cauchy sequence, ie for all \epsilon there is an N such that |x_{m}-x_{n}|<\epsilon for all n,m>N.

    Now |x_{m}-x_{n}|= |x_{m}-x_{m-1}+x_{m-1}-...+x_{n+1}-x_{n}| and the triangle inequality gives |x_{m}-x_{n}|\leq|x_{m}-x_{m-1}|+|x_{m-1}-x_{m-2}|+...+|x_{n+1}-x_{n}|.
    Then using the inequality given in the question I get |x_{m}-x_{n}|<\frac{1}{2^{m-1}}+\frac{1}{2^{m-2}}+...+\frac{1}{2^n}<\frac{1}{2^n}+\frac{1}{2^n}+  ...+\frac{1}{2^n}=\frac{m-n}{2^n}.
    This is where I get stuck, I think I need to get rid of that m-n somehow.
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  2. #2
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    Yes, you are quite close to the answer, right up to the inequality |x_{m}-x_{n}|< \frac{1}{2^{m-1}}+\frac{1}{2^{m-2}}+...+\frac{1}{2^n} (where m>n). Here, instead of estimating each individual term (replacing each term by the largest term 1/2^n), write the terms in the reverse order \frac{1}{2^n} + \frac{1}{2^{n+1}} + \frac{1}{2^{n+2}} + \ldots + \frac{1}{2^{m-1}}. This is a (finite) geometric series, whose sum is less than the infinite sum \sum_{r=n}^\infty \frac1{2^r} = \frac1{2^{n-1}}.
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  3. #3
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    Ah cheers, I never think of using geometric series!
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