Integers under Addition form Infinite Cyclic Group

Theorem
The additive group of integers $$\left({\mathbb{Z}, +}\right)$$ is an infinite cyclic group which is generated by the element $$1 \in \mathbb{Z}$$.

Proof
By Epimorphism from Integers to a Cyclic Group and integer multiplication:

$$\forall n \in \mathbb{Z}: n = +^n 1 \in \left \langle {1} \right \rangle$$

Thus $$\left({\mathbb{Z}, +}\right) = \left \langle {1} \right \rangle$$ and thus, by the definition of a cyclic group, is cyclic.