Positive Real Numbers Closed under Multiplication

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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 $a \times b \in \R_{\gt 0}$.

$\blacksquare$


Also see