Additive Group of Rationals is Normal Subgroup of Reals

Theorem
Let $$\left({\mathbb{Q}, +}\right)$$ be the Additive Group of Rational Numbers.

Let $$\left({\mathbb{R}, +}\right)$$ be the Additive Group of Real Numbers.

Then $$\left({\mathbb{Q}, +}\right)$$ is a normal subgroup of $$\left({\mathbb{R}, +}\right)$$.

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

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

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