Completeness of R2 with Taxicab Norm

[gtfitzpatrick]

Homework Statement

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

The Attempt at a Solution

you know that ,x_k \rightarrow x , y_k \rightarrow y

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

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

i've modified an answer from another question here, i think this work but I am not sure...

 

[Tinyboss]

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

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

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

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