Proof that the square root of any nonperfect square positive integer is irrational

Let us assume that $\sqrt{n}$ is rational where $n$ is a nonperfect square positive integer. Then it can be written as $\frac{p}{q}=n^2 \Rightarrow \frac{p^2}{q^2}=n$ \Rightarrow (q^2)n=p^2 $. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore$ \sqrt{n}$ is irrational.