Prove that there is infinite rational number between two rational number?

Let's take two rational numbers, 1 and 2.

Just in the middle, you have 3/2.
Between 1 and 3/2, you have 5/4 and between 3/2 and 2, you have 7/4.
Between 1 and 5/4, you have 9/8.
Then, you'll have 17/16, then 33/32, then 65/64, and so on, to infinity.

If we take the average of two rationals called p and q, then we have (p+q)/2

If we let p=a/b and q=c/d, then we get

Since ad+bc and 2bd are integers we have shown there is another rational between two rationals. This can be projected on and on infinitely.


EDIT: To would appear Unknown beat me... alas