Non-Zero Real Numbers Closed under Multiplication

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

Proof
We have that the real numbers form a field under the operations of addition and multiplication.

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

Thus, by definition, $\times$ is closed in $\left({\R_{\ne 0}, \times}\right)$.