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Proving a sequence converges

  1. Oct 13, 2015 #1
    1. The problem statement, all variables and given/known data: Prove that if xn is a sequence such that |xn - xn+1| ≤ (1/3n), for all n = 1,2,..., then it converges.


    2. Relevant equations: The definition of convergence.


    3. The attempt at a solution: I attempted to prove this by induction, so I am clearly far off the mark here. Any advice would be appreciated.
     
  2. jcsd
  3. Oct 13, 2015 #2

    andrewkirk

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    A good start would be to write down your definition of convergence. All the hints that are likely to be needed are in that definition.
     
  4. Oct 13, 2015 #3
    Our definition of convergence provided is the following:

    {an} converges to A if ∀ε>0, ∃a positive integer N such that n ≥ N => | an - A | < ε.

    I'm not sure how I am supposed to use this in the example above?
     
  5. Oct 13, 2015 #4

    Ray Vickson

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    Have you ever heard of Cauchy sequences?
     
  6. Oct 13, 2015 #5
    Yes.
     
  7. Oct 13, 2015 #6

    Ray Vickson

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    OK.... so?
     
  8. Oct 13, 2015 #7
    Do I just assume that xn can be substituted for the usual xn and that xn+1 be substituted for xm and attempt to find the n,n+1 ≥ N such that | xn - xn+1 | < ε, given ε > 0?
     
  9. Oct 13, 2015 #8

    Ray Vickson

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    No, that is not what a Cauchy sequence is all about. A sequence is Cauchy if, given any ##\epsilon > 0## there is an ##N = N(\epsilon)## such that ##|x_n - x_m|< \epsilon## for all ##n,m > N##. Note that this is ##|x_n - x_m|##, not just ##|x_n - x_{n+1}|##.

    You might guess that if I mention Cauchy sequences, that must be for a good reason.
     
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