Ok I’m not sure if this strictly constitutes a proof but here is one way of showing that there are infinitely many fractions by creating a one-one correspondence with the integers.

To create a one-one correspondence between a given set and the integers the first step is to list all the members of that set.

This is slightly more difficult with fractions than other simple sets because if you can’t example list them in order of magnitude because no matter which fraction you chose there is always one half its size. However it is possible by defining the following function :

For a give fraction x (in it’s lowest terms) where x=a/b

You can use this function to assign an integer value to all fractions.

You can now list all fractions, first in order of their values of f(x) and then, for those values corresponding to more than one fraction, in order of the denominators of the fractions. This will give an ordered list containing all fractions in existence so you easily create a one-one correspondence with the integers.

This shows that although there are in infinite number of fractions between every pair of integers there in fact no more fractions than integers.