Show if $x \in \mathbb{Q}$ and $x^2 \in \mathbb{Z}$, then $x \in \mathbb{Z}$.

Question

I proved by showing if $x \in \mathbb{Q}$ and $x \not\in \mathbb{Z}$, then $x^2 \not\in \mathbb{Z}$.

Because $x = \frac{m}{n}$, at least one of $m$, $n$ is odd.
$\text{odd}^2 = \text{odd}$.
Therefore $x^2 = \frac{m^2}{n^2}$ is also rational.
Therefore $x^2$ is not integer.
Proved by contrapositive.

Is my proof correct?

Thank you!

Answer 1

There is a gap in your proof, because saying that $x^2=m^2/n^2$ is also rational, does not immediately imply that $x^2$ is not integer. You need to give a further argument here. The proof has been given often here on MSE, see for example here.