Additive Group of Integers is Subgroup of Reals

Theorem
Let $$\left({\mathbb{Z}, +}\right)$$ be the Additive Group of Integers.

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

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

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

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

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

As the Additive Group of Real Numbers is abelian, from All Subgroups of Abelian Group are Normal it follows that $$\left({\mathbb{Z}, +}\right) \triangleleft \left({\mathbb{R}, +}\right)$$.