Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers

Examples of Order of Group Element
Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

The order of $2$ in $\struct {\R_{\ne 0}, \times}$ is infinite.

Proof
From Real Multiplication Identity is One, the identity of $\struct {\R_{\ne 0}, \times}$ is $1$.

There exists no $n \in \Z_{\ge 0}$ such that $2^n = 1$.

Hence the result by definition of infinite order element.