A question my geometry teacher could not answer - If two points determine distance, and two points on a line can always be divided by another point, then are there an infinite number of points within a finite space? Assuming/expecting that a 'point' inhabits some measurable amount of space, how can an infinite number of points exist within a finite space?

A question my geometry teacher could not answer - If two points determine distance, and two points on a line can always be divided by another point, then are there an infinite number of points within a finite space? Assuming/expecting that a 'point' inhabits some measurable amount of space, how can an infinite number of points exist within a finite space?