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 $\displaystyle x$ and $\displaystyle y$ are 2 rational numbers such that $\displaystyle x<y$.

Since $\displaystyle x$ and $\displaystyle y$ are rational

Therefore $\displaystyle \exists_a, _b, _c, _d\in Z $ with $\displaystyle b $ and $\displaystyle d $ not = 0 such that $\displaystyle x = \frac{a}{b}$ and $\displaystyle y = \frac{c}{d}$

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: $\displaystyle \sqrt{xy}$ 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.