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About square summable sequences space

  1. Jun 16, 2009 #1
    First, I'm sorry for my bad english.
    1. The problem statement, all variables and given/known data
    I need to disprove:
    [tex](x_n) \in \ell^2[/tex] is a Cauchy sequence, if [tex]\displaystyle \lim_{x \to \infty} d(x_n, x_{n+1})=0[/tex].
    2. Relevant equations
    Ok, sequence is Cauchy sequence if [tex] \exists n_0 \; \forall p,q>0 \; d(x_p,x_q) \rightarrow 0 [/tex]


    3. The attempt at a solution
    Has someone idea about this? I tried 1/ln(x) and many examples like this one, but all this are wrong.
     
  2. jcsd
  3. Jun 16, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You mean [tex]\forall p,q> n_0[/tex]

    Here's a hint: [tex]\sum_{n=1}^\infty\frac{1}{n}[/tex] does not converge.

    Your English is excellent. (Well, except for not capitalizing "English"!)
     
  4. Jun 16, 2009 #3
    Of course, [tex]x_n=(1, \frac{1}{2}, ... , \frac{1}{\sqrt{n}})[/tex] is what I'm looking for. And this sequence is square summable because [tex]x_n[/tex] is finite.

    Thanks! ^_^
     
    Last edited: Jun 16, 2009
  5. Jun 16, 2009 #4
    No worries mate, Latvians speak English well.
     
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