Prove that if $x \in \mathbb{R}$ and $y \in \mathbb{R}$, then $xy \leq |xy|$.

Question

Prove that if $x \in \mathbb{R}$ and $y \in \mathbb{R}$, then $xy \leq |xy|$.

Proof (Attempt):

Case $1$ : $x \geq 0$ and $y \geq 0$. Then $xy \geq 0$ so $xy =|xy|$.

Case $2$ : $x \geq 0$ and $y < 0$. Then $xy \leq 0$ so $xy \leq |xy|$.

Case $3$ : $x < 0$ and $y \geq 0$. Then $xy \leq 0$ so $xy \leq |xy|$.

Case $4$ : $x < 0$ and $y < 0$. Then $xy > 0$ so $xy = |xy|$.

This is what I came up with, though I wonder if I should concern the case where either one of them is zero separately as another case. My question is, does this proof seem okay or should I include sub cases to account for when $x$ or $y$ is zero?

Answer 1

The simplest is to use one of the possible definitions of the absolute value on $\mathbf R$: $$\lvert a\rvert=\max(a, -a).$$