• Apr 1st 2008, 10:38 AM
patricia-donnelly
Hello

I have a question that I've come across and I'm having trouble with it. Any light anyone could shed on it would be an enormous help. The question is

(a)Show that a sequence (x(n)) where x(n) = (xn1, xn2) converges to x = (x1, x2) in R2 with the sup metric if and only if xn1 → x1 and xn2→ x2 in R.
(b)Prove that R2 with sup metric is complete

Any help would be appreciated
Thanks
• Apr 1st 2008, 12:15 PM
Opalg
Quote:

Originally Posted by patricia-donnelly
Hello

I have a question that I've come across and I'm having trouble with it. Any light anyone could shed on it would be an enormous help. The question is

(a)Show that a sequence (x(n)) where x(n) = (xn1, xn2) converges to x = (x1, x2) in R2 with the sup metric if and only if xn1 → x1 and xn2→ x2 in R.
(b)Prove that R2 with sup metric is complete

Any help would be appreciated
Thanks

If $\displaystyle \mathbf{x}=(x_1,x_2)$ and $\displaystyle \mathbf{y}=(y_1,y_2)$ are points in R^2 then the distance between them in the sup metric is given by $\displaystyle d(\mathbf{x},\mathbf{y}) = \max\{|x_1-y_1|,|x_2-y_2|\}$. Thus given ε>0, a necessary and sufficient conditon for $\displaystyle d(\mathbf{x},\mathbf{y})<\epsilon$ is that $\displaystyle |x_1-y_1|<\epsilon$ and $\displaystyle |x_2-y_2|<\epsilon$. That is the main thing that you need to use in order to prove both parts of this question.