Proof that between two rational numbers we can include an irrational number or Irrational numbers are dense in rational

Take two different rational numbers $r_1,r_2$, w.l.o.g we can assume that $r_2 > r_1 \Rightarrow r_2-r_1 > 0$. Now we choose an irrational number between $0$ and $1$. Before we do that we must recall that reciprocal of an irrational number is an irrational number ie. $\frac{1}{\text{irrational}}= \text{irrational}$ and rational + irrational = irrational. There are many choices of irrational numbers to start with lets start with $\frac{1}{\sqrt{5}}$. $\Rightarrow 0\frac{1}{\sqrt{5}}$ Multiplyting all the sides by a positive number $r_2-r_1$ we get $0 < (r_2-r_1)\frac{1}{\sqrt{5}} < r_2-_1$ Adding $r_1$ to all the sides we get $r_1 < r_1+\frac{r_2-r_1}{\sqrt{5}}< r_2$. As $r_1+\frac{r_2-r_1}{\sqrt{5}}$ is irrational numbers between $r_1$ and $r_2$.