Given $X$ enumerable, prove that always exist a metric in which every $p\in X$ is a limit point

Question

That's it. Given a infinite enumerable set $X$ show that we can define a metric in $X$ in which every point of $X$ is limit point.

I have no idea what to do here. I can show a metric in which every point is isolated, but not limit point... and I have no clue on how to proceed, so if you just give me a hint but not the exact answer I'd be grateful too!

Answer 1

There is a $1-1$ map $T$ from $X$ to the rational number on $[0,1]$. The ordinary distance $d$ on $[0,1]$ can be transformed the metric on $X$, that is D(x,y)=d(Tx,Ty), $\forall x,y\in X$.