Let the two rational numbers be (a) and (b), with (a < b).
Find a common denominator for (a) and (b).
Rewrite both numbers with the same denominator.
Choose any fraction whose numerator lies strictly between the two numerators.
If no integer lies between the numerators, multiply both numbers by a larger common denominator to create more space.
Take the average (frac{a+b}{2}), which is always a rational number between (a) and (b).
Repeat the averaging process to find more rational numbers between them.
Use the fact that between any two rational numbers, there are infinitely many rational numbers.