All proofs is geometry depend on the particular set of definitions and axioms in use on the textbook. From the wording of this question it seems as if you have what Edwin Moise calls theRuler Axiom: Every line has a coordinate system.

So yes you are correct that this means there is a one-to-one correspondence between the points on a line and the real numbers.

Note that between any two numbers that is a third number: .

This means that between any two points on a line there is a third point.