Additive Group of Integers is Subgroup of Reals

Theorem
Let $\left({\Z, +}\right)$ be the additive group of integers.

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

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

Proof
From Additive Group of Integers is Subgroup of Rationals, $\left({\Z, +}\right) \lhd \left({\Q, +}\right)$.

From Additive Group of Rationals is Subgroup of Reals, $\left({\Q, +}\right) \lhd \left({\R, +}\right)$.

Thus $\left({\Z, +}\right) \le \left({\R, +}\right)$.

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