Integers under Addition form Infinite Cyclic Group

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

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


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

Thus:
 * $\left({\Z, +}\right) = \left \langle {1} \right \rangle$

and thus, by the definition of a cyclic group, is cyclic.