- #1
gtfitzpatrick
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Homework Statement
Given R is complete, prove that R2 is complete with the taxicab norm
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 I am not sure...