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?
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?
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