Suppose $x$ and $y$ are natural numbers. Show that $xy$ odd implies that $x$ and $y$ are both odd.

Question

Is my following proof correct using the contrapositive method?

Contrapositive Statement:

Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even.

Proof:

For $x,y\in \mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $m\in \mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=my\in \mathbb{N}$. Thus, $xy$ is even.

Answer 1

Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.

It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.