proving of properties of the modulus function by exhaustion - $\forall\ x \in\mathbb{ R} : |xy| = |x||y|$

just need clarification with a quick proof of properties of the modulus function to make sure I'm doing the right thing.

$$\forall\ x \in\mathbb{R} : |xy| = |x||y|$$

If I let $x,y \in\mathbb{ R}$ and prove by exhaustion do I do this by considering when $x \ge 0, y\ge0$ then $|xy| = |x||y|$ and when $x < 0, y<0$ then $|xy| = |-x||-y| = |x||y|$.

Is this the correct way to go about it?

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That works, but then to complete the proof, you also need to consider,

• the cases where $\;x \geq 0, \;y\leq 0,\;$ and by symmetry: $\;x \leq 0,\; y\geq 0$.

Since there are only four cases to consider, proof by exhaustion works just fine.

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Just for the sake of pleasing the most pedantic among us, the proof will look nicer if you write it like this: Assume $x\ge 0$ and $y\ge 0$. Then $xy\ge 0$, and thus, by definition, $|x|=x$ and $|y|=y$ and $|xy|=xy$. So, $|xy|=xy=|x||y|$.