I know that every countable metric space is compact metric space and Every compact metric space is totally bounded and complete. That is for any $s$ there is an $s-$net(A subset of $X$). That is for any $s$, for each $x\in X$ there is $p\in$ $s-$net such that $d(p,x)<s$. I don't understand how the underlined statement is coming. please explain.
This is by Zorn's lemma. Consider the poset $P$ of all subsets $A\subseteq X$ such that $d(a,b)\geq s$ for all $a,b\in A$, ordered by inclusion. A union of chain of elements of $P$ is again an element of $P$, so every chain in $P$ has an upper bound. By Zorn's lemma, $P$ has a maximal element.