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If we define a metric $d : \mathbb{Z^+} \times \mathbb{Z^+} \to \mathbb{R}$, where $\mathbb{Z^+}$ is the set of positive integers and $\mathbb{R}$ is the set of reals, as follows:

$$d(m,n) := |m^{-1} - n^{-1}|$$

for all $m$, $n \in \mathbb{Z^+}$, then how to prove if this metric space is complete?

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I think you mean $d \colon \mathbf{Z^+}\times \mathbf{Z^+} \to \mathbf{R^+}$. As with your other question, a metric takes pairs of points in a space to a distance. –  Ross Millikan Jan 28 '13 at 0:38
    
Oops sorry! It's the set of positive integers. –  Saaqib Mahmuud Jan 28 '13 at 0:40
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Hint: What's the limit of $a_n = n$? Is it a positive integer?

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