Q:

Set of all positive integers with metric $d(m,n) = \mid \frac{1}{m} - \frac{1}{n} \mid$ is not complete.

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

A:

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.

