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