Characterization of countable groups as groups with a left-invariant distance with finite balls

Question

In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, the following characterization of countable groups: $G$ is countable if and only if it has a left-invariant metric with finite balls. See Proposition 1.A.1. and Lemma 2.B.5 of https://arxiv.org/pdf/1403.3796.pdf ,

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, so the proof given in the book is unnecessarily too complicated for my purposes.

Answer 1

I have not found a proof in the book. But the statement is not difficult. Assign to each element of a countable group $G$ a natural number $0,1,2,...$ (for every $n$ only finite number of elements $g$ are assigned the same numbers $n(g)=n$, $n(1)=0$ and 1 is the only element with number 0, $n(g^{-1})=n(g)$). Then define a norm $|g|$ of $g$ as the smallest sum $n(g_1)+...+n(g_k)$ such that $g=g_1g_2...g_k$ in $G$. Then define the distance $dist(g,h)=|g^{-1}h|$. It is left invariant and has finite balls.