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Math Help - Complete metric spaces.

  1. #1
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    Complete metric spaces.

    If completeness of a metric space is defined as,

    A metric space (M, d) is complete if every Cauchy sequence {x_n} in M converges.

    I am stuck trying to verify the following metric space is complete.
    \mathbb{R}^k with the metric defined as d(x,y)=\max _{1\leqslant i\leqslant k}\left|x_i-y_i\right|

    Cleary this is a metric space, but how can I see if its complete?

    I thought a verification of completeness might be similar to testing whether the euclidean space is compete (that is with aid of the Bolzano Weierstrass theorem). But that theorem doesn't hold for general metric spaces.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by aukie View Post
    If completeness of a metric space is defined as,

    A metric space (M, d) is complete if every Cauchy sequence {x_n} in M converges.

    I am stuck trying to verify the following metric space is complete.
    \mathbb{R}^k with the metric defined as d(x,y)=\max _{1\leqslant i\leqslant k}\left|x_i-y_i\right|

    Cleary this is a metric space, but how can I see if its complete?

    I thought a verification of completeness might be similar to testing whether the euclidean space is compete (that is with aid of the Bolzano Weierstrass theorem). But that theorem doesn't hold for general metric spaces.
    Let \{x_n\}\subset\mathbb{R}^k be a Cauchy sequence. For each x_j\in\{x_n\}, define x_j^i, 1\leq i\leq k, to be the ith component of x_j.

    Prove that for each i the sequences \{x_n^i\} are all Cauchy under the standard metric over \mathbb{R} and are therefore convergent to numbers y_1,...,y_k\in\mathbb{R}. Then show that the sequence \{x_n\} converges to y=(y_1,...,y_k).
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