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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. There is no proof in the book.

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 descrete.

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.

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.

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. There is no proof in the book.

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 descrete.

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.

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.