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Cauchy sequences homework

  1. Feb 21, 2010 #1
    1. The problem statement, all variables and given/known data

    Given R is complete, prove that R2 is complete with the taxicab norm

    3. The attempt at a solution

    you know that ,xk [tex]\rightarrow[/tex] x , yk [tex]\rightarrow[/tex] y

    Then, given [tex]\epsilon[/tex], choose Nx and Ny so that [tex]\left|x_n - x_m\left|[/tex] and [tex]\left|y_n - y_m\left|[/tex] are less than [tex]\epsilon/2[/tex] respectively, whenever m,n [tex]\geq[/tex] N = [tex]\left|N_x\left|+\left|N_y\left|[/tex].

    Then d(([tex]\ x_n,y_n[/tex]),([tex]\ x_m,y_m[/tex])) = [tex]\sqrt{(x_n - x_m)^2 + (y_n - y_m)^2}[/tex] [tex]\leq[/tex] [tex]\sqrt{(\epsilon^2 /4) + (\epsilon^2 /4)}[/tex] = [tex]\epsilon/2[/tex] < [tex]\epsilon[/tex]

    i've modified an answer from another question here, i think this work but im not sure...
     
  2. jcsd
  3. Feb 23, 2010 #2
    Re: Completness

    Let [tex]x_n,y_n[/tex] be Cauchy sequences in R, then you know they have limits, x,y, elements of R.

    Given epsilon>0, choose Nx such that [tex]|x_n-x|<\varepsilon/2[/tex] for all n>Nx, and choose Ny the same way. Then let N=max(Nx,Ny).

    Then for n>N, you have:
    [tex]d((x_n,y_n),(x,y))=|x_n-x|+|y_n-y|<\varepsilon.[/tex]
    (Remember it specified the taxicab norm, not Euclidean!)

    Now you know that [tex](x_n,y_n)[/tex] converges to (x,y), and that (x,y) is actually in R2. So R2 is complete.
     
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