Let $X$ be the integers with metric $ρ(m,n)=1$, except that $ρ(n,n)=0$. Check that $ρ$ is a metric. Show that $X$ is closed and bounded, but not compact.
This is a "made-up" example demonstrating closed and bounded doesn't imply compactess in more general metric space. I checked that $\rho$ is a metric already. Yet I have no idea how to approach "showing $X$ is closed and bounded." I visualized this metric to be a set of number with just $0$ and $1$ (or maybe this is not correct?). Also, I doubt that this metric is NOT compact. Anyway, I'd appreciate if you can help! Meanwhile, do we use a ball $B(0,1)$ in general to show that a metric is closed and bounded? If so, why?