[math-fun] iterated insertion of mediants

If you start with two arbitrary (positive) rational numbers p,q and start iteratively interpolating mediants, and you reduce every unreduced fraction you encounter to lowest terms, do you get all the rationals between p and q? Example: Start with 1/2 and 3/1. 1/2 3/1 gives 1/2 4/3 3/1 gives 1/2 1/1 4/3 7/4 3/1 gives 1/2 2/3 1/1 5/4 4/3 11/7 7/4 2/1 3/1 etc. After n rounds of mediant interpolation, you have 2^n+1 fractions. Jim