Question 442256
Keep in mind that if x and y are rational, then *[tex \frac{x+y}{2}] is also a rational number. The proof is quite simple; simply let *[tex x = \frac{a}{b}] and *[tex y = \frac{c}{d}], where a,b,c,d are integers and b and d are nonzero. Then

*[tex \LARGE \frac{x}{y} = \frac{ \frac{a}{b} + \frac{c}{d}} {2} = \frac{ad + bc}{2}]. This is a ratio of two integers, so the average of x and y is a rational number.

Hence, if we let *[tex z_1 = y], we may let *[tex z_2 = \frac{x + z_1}{2}], and in general, *[tex z_{i+1} = \frac{x + z_i}{2}]. All of these numbers are rational, and we can go on indefinitely, so there must be infinitely many rational numbers, regardless of how far x and y are.