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Convergence of Sequence Does {An^2} converges => {An} converges? How to prove it?

  1. Sep 8, 2010 #1
    Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove it?"

    Does sequence {An^2} converges implies to sequence {An} converges? True or False. How to prove it?

    I kinda think it is false, but couldn’t think of any counterexample to directly proof it. So I try to use the 1) definition of convergence and 2) the Comparison Lemma to prove it, but kinda stucked.

    Proof1: (Use definition of convergence)
    Let sequence An^2 converges to a^2
    Then according to the definition of convergence
    For every E>0, Find N such that
    |An^2-a^2|<E for all n>N
    |(An-a)(An+a)|=|An-a||An+a|<E
    |An-a|<E/|An+a|)

    How can I go from here?

    So if I can let N=E/|An+a|+1, then An converges to a. But I can’t define N that has a sequence in it, can I?

    Comparison Lemma states “Let sequence {An} converges to a, and let {Bn} be a sequence such that |Bn-b|<= C|An-a| for some C>0, then Bn converges to b”

    Proof2: (Use Comparison Lemma)
    Let sequence An^2 converges to a^2
    Let {An} be a sequence such that |An-a|<=C|An^2-a^2| for some C>0
    Now I need to find an C>0 so that I can prove that sequence An converges to a

    How can I go from here?
     
  2. jcsd
  3. Sep 8, 2010 #2

    mathman

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    Re: Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove i

    Your first proof looks OK unless a=0.
     
  4. Sep 8, 2010 #3
    Re: Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove i

    1, -1, 1, -1, 1, -1, ...
     
  5. Sep 8, 2010 #4

    statdad

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    Re: Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove i

    :approve:
     
  6. Sep 8, 2010 #5
    Re: Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove i

    So choose sequence An^2 = [(-1)^n]^2 and the sequence An^2 converges to 1, but An is NOT convergent (divergent) sequence. Super! Thanks!
     
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