Additive Group of Integers is Subgroup of Reals

Theorem
Let $\struct {\Z, +}$ be the additive group of integers.

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

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

Proof
From Additive Group of Integers is Subgroup of Rationals, $\struct {\Z, +} \lhd \struct {\Q, +}$.

From Additive Group of Rationals is Subgroup of Reals, $\struct {\Q, +} \lhd \struct {\R, +}$.

Thus $\struct {\Z, +} \le \struct {\R, +}$.

As the additive group of real numbers is abelian, from Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +} \lhd \struct {\R, +}$.