Element of Cyclic Group is not necessarily Generator

Theorem
Let $\gen g = G$ be a cyclic group.

Let $a \in G$

Then it is not necessarily the case that $a$ is also a generator of $G$.

Proof
Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then $4$ is not a generator of $\gen 2$.