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Math Help - Find an "interesting" Cauchy sequence.

  1. #1
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    Find an "interesting" Cauchy sequence.

    For a rational number x \neq 0, write it as 2^k\frac{m}{n}, where m and n are integers that have no common factors, and let |x|_{2} = 2^{-k}. Define |0|_{2} = 0. Let d(x,y) = |x-y|_{2}.

    Prove d(x,z) \leq d(x,y) + d(y,z) for all rational x, y, z.

    Now since d(x,y) = d(y,x) and d(x,y) \geq 0 with equality only if x = y. Write an interesting Cauchy sequence in this metric.
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  2. #2
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    Quote Originally Posted by cgiulz View Post
    For a rational number x \neq 0, write it as 2^k\frac{m}{n}, where m and n are integers that have no common factors, and let |x|_{2} = 2^{-k}. Define |0|_{2} = 0. Let d(x,y) = |x-y|_{2}.

    Prove d(x,z) \leq d(x,y) + d(y,z) for all rational x, y, z.

    Now since d(x,y) = d(y,x) and d(x,y) \geq 0 with equality only if x = y. Write an interesting Cauchy sequence in this metric.

    For positive integers m,n we have that m > n <==> 2^(-m) < 2^(-n) , so

    |x + y|_2 <= max{|x|_2 , |y|_2} <= |x|_2 + |y|_2 and from here the

    triangle inequality follows.

    As for an interesting Cauchy sequence: what about the sequence

    {2^n} = {2, 4, 8, 16,...} ?

    Tonio
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