Positive Real Numbers Closed under Multiplication

Theorem

The set $\R_{>0}$ of strictly positive real numbers is closed under multiplication:

$\forall a, b \in \R_{\gt 0}: a \times b \in \R_{\gt 0}$

Proof

Let $a, b \in \R_{\gt 0}$

It is seen that the Real Numbers form Ordered Integral Domain.

It then follows from Positive Elements of Ordered Ring that $a \times b \in \R_{\gt 0}$.

$\blacksquare$