1. The problem statement, all variables and given/known data Prove that a line in a metric geometry has infinitely many points. 2. The attempt at a solution I can't use any real analysis, like completeness. I can only use geometry to prove this, specifically distances and rulers. Intituvely I understand why. Any segment with at least two points has infinitely many points, because, intuitively, given any two distinct points, there is a third one, distinct from both of them and so on. But how can I prove this formally?