In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, without proof, the following characterization of countable groups: $G$ is countable if and only if it has a left-invariant metric with finite balls.See page 10 of https://arxiv.org/pdf/1403.3796.pdf , Proposition 1.A.1.

I can easily see that if a group admits a left invariant metric such that every ball of finite radius is finite, then $G$ is countable. But what about the converse, how can we obtain a metric on $G$ with the above properties if $G$ is countable?

I am interested in the case when $G$ is discrete.