I need to prove using the method of proof by construction that: Between every two distinct rational numbers lies an irrational number.
I started my proof assuming that and are 2 rational numbers such that .
Since and are rational
Therefore with and not = 0 such that and
I need to construct an element q that can be between x and y and that has the properties of an irrational number. I thought about using the geometric mean of the numbers x and y: but the geometric mean can be a rational number or an irrational number. I don't know if I have to prove this element q to be irrational all the time.
Any assistance would be greatly appreciated, thank you.