Multiplicative Group of Rationals is Normal Subgroup of Reals

Theorem
Let $$\left({\Q^*, \times}\right)$$ be the Multiplicative Group of Rational Numbers.

Let $$\left({\R^*, \times}\right)$$ be the Multiplicative Group of Real Numbers.

Then $$\left({\Q^*, \times}\right)$$ is a normal subgroup of $$\left({\R^*, \times}\right)$$.

Proof
From the definition of real numbers, it is clear that $$\Q$$ is a subset of $$\R$$.

As $$\left({\R^*, \times}\right)$$ is a group, and $$\left({\Q^*, \times}\right)$$ is a group, it follows from the definition of subgroup that $$\left({\Q^*, \times}\right)$$ is a subgroup of $$\left({\R^*, \times}\right)$$.

As $$\left({\R^*, \times}\right)$$ is abelian, it follows from All Subgroups of Abelian Group are Normal that $$\left({\Q^*, \times}\right)$$ is normal in $$\left({\R^*, \times}\right)$$.