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

I can see easily that if the group admits 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.