Proving that every countable metric space is disconnected? Any hints on how I should go about this? Thanks.

The statement is false if $|X|=1$, so countable here is probably supposed to be understood as countably infinite. 
HINT: Let $X$ be the space, and let $x,y\in X$ with $x\ne y$. Let $r=d(x,y)>0$. Use the fact that $X$ is countable to show that there must be an $s\in(0,r)$ such that 
$$\{z\in X:d(x,z)=s\}=\varnothing\;.$$
If $B(x,s)=\{z\in X:d(x,z)<s\}$, show that $B(x,s)$ and $X\setminus B(x,s)$ are non-empty open sets in $X$, and conclude that $X$ is not connected.
