Can anyone confirm if my proof is correct, please?
Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.”
Proof:-
Assume that both $x$ and $y$ are rational.
Now, let $x = \dfrac pq$ and $y = \dfrac mn$ since both of them are rational.
$xy =\dfrac pq * \dfrac mn = \dfrac{pm}{qn}$
Thus, if the product $xy$ can be written as a fraction, it's not a irrational number.
Therefore if one of $x$ and $y$ is not irrational, then the product is not irrational.
By the principle of proof by contraposition, If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.
Best Answer
I would specify that $p,q,m,n\in\mathbb{Z},q\neq0,n\neq0$, in your first line of the proof.
Other than that, the rest looks good.