Non-Zero Real Numbers Closed under Multiplication/Proof 1

Theorem

The set of non-zero real numbers is closed under multiplication:

$\forall x, y \in \R_{\ne 0}: x \times y \in \R_{\ne 0}$

Proof

Recall that Real Numbers form Field under the operations of addition and multiplication.

By definition of a field, the algebraic structure $\struct {\R_{\ne 0}, \times}$ is a group.

Thus, by definition, $\times$ is closed in $\struct {\R_{\ne 0}, \times}$.

$\blacksquare$