Prove the following For any rational numbers x and y if x y

Prove the following: For any rational numbers x and y, if x < y, then there is a rational number z such that

x < z < y

Solution

let z=(x+y)/2

we will now show that z lies between x and y.

clearly, (y-z) = (y-x)/2 > 0,

hencez<y.

Similarly, (z-x) = (y-x)/2, which implies x<z.

Hence, Proved.