If $a,b\in\mathbb R_{\ge0}$, then $ab\in\mathbb R_{\ge0}$

Question

The Second Order Property of $\mathbb R$ states that:

If $a,b\in\mathbb R^+$, then $ab\in\mathbb R^+$

However, is the following also true?

If $a,b\in\mathbb R_{\ge0}$, then $ab\in\mathbb R_{\ge0}$

Intuitively, it certainly seems to be true. And if it indeed is true, how to prove it?

Answer 1

$\mathbb R_{\ge0}$ partitions into two subsets, $\mathbb R^+$ and $\{0\}$. Each of $a$ and $b$ is in exactly one of these subsets.

If $a,b\in\mathbb R^+$, then by the given property $ab\in\mathbb R^+\subset\mathbb R_{\ge0}$. If $a\in\{0\}$, $a=0$ and thus $ab=0\in\mathbb R_{\ge0}$, and similarly for $b\in\{0\}$.