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Math Help - advanced algebra

  1. #1
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    advanced algebra

    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
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  2. #2
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    Quote Originally Posted by patricia-donnelly View Post
    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.
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