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Convergence and Cauchy Criterion

  1. Sep 24, 2015 #1
    1. The problem statement, all variables and given/known data
    Suppose the sequence (xn) satisfies |xn + 1 - xn| < 1/n2, prove that (xn) is convergent.

    2. Relevant equations
    |xn - xm| < ɛ

    3. The attempt at a solution
    If m > n, then
    |xn - xm|
    < |xn - xn + 1| + |xn + 1 - xn + 2| + ... + |xm - 1 - xm|
    < 1/n2 + 1/(n+1)2 + ... + 1/(m - 1)2
    < [1/(n - 1) - 1/n] + [1/n - 1/(n + 1)] + ... + [1/(m - 2) - 1/(m - 1)] = 1/(n - 1) - 1/(m - 1)
    < 1/(n - 1)

    Let ɛ be given. Choose m > n > N := [1/ɛ] + 1 such that |xn - xm| < ɛ for all m > n > N.

    Is there any problem with my proof?
     
  2. jcsd
  3. Sep 24, 2015 #2

    Fredrik

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    Gold Member

    The calculations look good, but the statements surrounding it do not. In particular:
    1. It doesn't make sense to say that you choose m such that some statement is true for all m.
    2. You didn't say that ##\varepsilon## is a positive real number, so the reader can wonder how you intend to make ##|x_n-x_m|<\varepsilon## when ##\varepsilon=-1##.
    3. The calculation didn't actually conclude that ##|x_n-x_m|<\varepsilon##.
    Edit: 4. If the [x] notation means what I think it does, your definition of N doesn't work.
     
    Last edited: Sep 24, 2015
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