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Math Help - Prove Or Disprove Completeness of Z

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    Prove Or Disprove Completeness of Z

    For this problem, we're dealing with the set of integers with the metric d(x,y)=\frac{1}{5^k} where k is the highest power of of 5 that divides x-y.

    I'm trying to either prove or disprove that this is a complete space. I know I have to either come up with a Cauchy sequence of integers that does not converge in the integers with respect to d, or show that every Cauchy sequence in the integers converges in the integers with respect to d. I'm just not seeing which one is the case. Does anyone have a hint to get me going in the right direction?
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    Quote Originally Posted by mathematicalbagpiper View Post
    For this problem, we're dealing with the set of integers with the metric d(x,y)=\frac{1}{5^k} where k is the highest power of of 5 that divides x-y.

    I'm trying to either prove or disprove that this is a complete space. I know I have to either come up with a Cauchy sequence of integers that does not converge in the integers with respect to d, or show that every Cauchy sequence in the integers converges in the integers with respect to d. I'm just not seeing which one is the case. Does anyone have a hint to get me going in the right direction?
    Think about the sequence (x_n), where x_n = 5^n. Then read about p-adic numbers (with p=5 here).
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