Additive Group of Rationals is Normal Subgroup of Reals

Theorem
Let $\left({\Q, +}\right)$ be the additive group of rational numbers.

Let $\left({\R, +}\right)$ be the additive group of real numbers.

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

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

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

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