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Math Help - contractive sequence

  1. #1
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    contractive sequence

    If  x_n is a sequence and there is a  c \geq 1 such that  |x_{k+1}-x_{k}| > c|x_{k}-x_{k-1}| for all  k > 1 , then can  x_n converge?

    We claim that if  n \in \bold{N} , then  |x_{k}-x_{k+1}| > c^{k-1}|x_{1}-x_{2}| . For  k = 1 ,  |x_{1}-x_{2}| > |x_{1}-x_{2}| , which is false. So it cannot be a contractive sequence?
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  2. #2
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    although this seems too simple.
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  3. #3
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    Quote Originally Posted by particlejohn View Post
    If  x_n is a sequence and there is a  c \geq 1 such that  |x_{k+1}-x_{k}| > c|x_{k}-x_{k-1}| for all  k > 1 , then can  x_n converge?
    Note that x_1 \not = x_2. Because if this was not the case then 0 = |x_2-x_1| > c|x_1 - x_0| which is impossible. But as you showed |x_{n+1} - x_n| > c^{n-1}|x_1-x_2| \geq |x_1-x_2|. Let 0<\epsilon < |x_1-x_2|. Then |x_{n+1} - x_n| > \epsilon for n\geq 2. Therfore the sequence \{x_n\} is not Cauchy and therefore not convergent.
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