advanced algebra

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