Prove that $\sqrt m$ is irrational by showing that the set $\{n\in\mathbb N: n\sqrt m\in\mathbb N\}$ is empty

Question

Let $m\in\mathbb N$ be such that $m\neq k^2$ for all $k\in\mathbb N$. Prove that $\sqrt m$ is irrational by showing that the set $\{n\in\mathbb N: n\sqrt m\in\mathbb N\}$ must be empty.

Answer 1

Hint: If there were such an $n$, then $(n\sqrt{m})^2=n^2m$ would be the square of an integer. (But $m$ shouldn't be writable as $k^2$ for any $k$)