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I am reading kreyszig functional analysis book where I got this problem:

Let $X$ be the set of all positive integers and $d(m,n) = \mid \frac{1}{m} - \frac{1}{n} \mid$. I have to show that $(X,d)$ is not complete metric space.

I took sequence $(x_n) = (n)$ which I showed that cauchy in $X$. But, I am not sure whether I am correct or not. I am also struggling with showing that this sequence is not convergent in $X$.

Thanks for helping me

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What could it converge to? Use proof by contradiction – Thomas Andrews Jun 1 '12 at 3:19
@JonasMeyer I will remember that. I have to edit now. – srijan Jun 1 '12 at 3:54
up vote 1 down vote accepted

I think you are on the right track. You want to show that your sequence does not converge. If it did converge, it would have a limit $L$. So say "Suppose the sequence converges. Then it has a limit $L$ such that… (definition of limit)." Then prove a contradiction, which shows that no such limit $L$ can exist.

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Suppose it has limit $L$ then $\lim _{n\rightarrow \infty}d(n, L) =\mid \frac{1}{L}\mid $. Now what argument should I give in support of our claim? – srijan Jun 1 '12 at 3:35
@srijan: That's good: For each positive integer $L$, $d(n,L)\to \frac{1}{L}$, and $\frac{1}{L}\neq 0$, so this shows that... – Jonas Meyer Jun 1 '12 at 3:39
@JonasMeyer Thanks to both of you. – srijan Jun 1 '12 at 3:53

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