>>11362970This proves there are always infinite real numbers between any two nonequal reals. For the special case of x and y rational, it proves infinite rationals between them.
What it doesn't prove is infinite rationals between any two reals, since the average of two reals isn't necessarily rational. However, a proof of this property (called the density of rationals in the reals) isn't complicated, although I'm too lazy to write it out.
Basically, given two nonequal reals, there's a nonzero finite distance between them. You can construct a rational number by choosing a denominator n large enough so the spacing between m/n and (m+1)/n is less than the distance between the reals, which implies for some integer k, k/n is between the two reals.
