Additive Group of Rationals is Normal Subgroup of Reals

Theorem
Let $\struct {\Q, +}$ be the additive group of rational numbers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Then $\struct {\Q, +}$ is a normal subgroup of $\struct {\R, +}$.

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

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

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