Proof that $\sqrt{20}$ is irrational number validation

Question

$$\sqrt{20}=\frac{m}{n}$$ $$20n^2=m^2$$ $$m = 20q \implies 20n^2=400q^2$$ $$n^2 = 20q^2 \implies \frac{n^2}{m^2} = \frac{20q^2}{20q} \implies \text{Contradiction}$$

Is my proof correct ?