Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact.
I know that sequentially compact and compact are equivalent for metric spaces. So in order to show it is not compact I plan on showing it is not sequentially compact (meaning that not every sequence in $X$ has convergent subsequence). So I need to show that $(x_n)$ does not have a convergent subsequence, but how would I go about doing that? Or should I be attempting a completely different approach for my proof?