Positive Real Numbers Closed under Division

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Theorem

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

$\forall a, b \in \R_{>0}: a \div b \in \R_{>0}$


Proof

From the definition of division:

$a \div b := a \times \paren {\dfrac 1 b}$

where $\dfrac 1 b$ is the inverse for real number multiplication.

From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, the algebraic structure $\struct {\R_{>0}, \times}$ forms a group.

Thus it follows that:

$\forall a, b \in \R_{>0}: a \times \paren {\dfrac 1 b} \in \R$

Therefore real number division is closed in $\R_{>0}$.

$\blacksquare$


Sources