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Let $X$ be a set of all positive integers and define metric $d$ on $X$ by $d(m,n)=|m^{-1} - n^{-1}|$. I'm required to show $(X,d)$ is not a complete space.

SOLUTION:

Let $\{x_n\}$ be any Cauchy sequence in $X$. Then choose $\epsilon=0.5$; there exists a number $N$ such that d(x_n,x_m)<0.5 for all $m,n>N$.

d(x_n,x_m)=|x_n^{-1} - x_m^{-1}|<0.5.

I'm not sure if $\{x_n\}$ may converge here, I beg help please.

kind regards

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    A way of coming up with a counterexample here is to ask yourself the question: when are two integers $m$ and $n$ close to each other with respect to this metric? Obviously this happens, if $m=n$, but that's the case with every metric space. Another automatic way for the distance to be small in this case is, when both $m$ and $n$ are large (as then their reciprocals are small). There we have more elbow room, and a chance to come up with a Cauchy-sequence that does not have a limit.2012-04-14

2 Answers 2

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The sequence $x_n = n \in \mathbb{Z}$ is a Cauchy sequence for $d$. And $(x_n)$ don't converge to a limit.

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Using the definition of $d$, it is easy to verify that the sequence $x_n=(1,2,3,\ldots)$ is a Cauchy sequence in $X=Z^+$. We see that $\forall k \in X, d(x_n,k)\rightarrow k^{(-1)}$ as $n \rightarrow \infty$. Hence $(x_n)$ does not converge in $X$ and so $X$ is not complete.