Epimorphism from Integers to Cyclic Group

Theorem
Let $$\left \langle {a} \right \rangle = \left({G, \circ}\right)$$ be a cyclic group.

Let $$f: \mathbb{Z} \to G$$ be a mapping defined as:

$$\forall n \in \mathbb{Z}: f \left({n}\right) = a^n$$.

Then $$f$$ is an epimorphism from $$\left({\mathbb{Z}, +}\right)$$ onto $$\left \langle {a} \right \rangle$$.

Proof
By the induction, $$\forall n \in \mathbb{N}: a^n \in \left \langle {a} \right \rangle$$

Hence by the Index Law for Monoids, $$\forall n \in \mathbb{Z}: a^n \in \left \langle {a} \right \rangle$$.

Also, by Index Law for Sum of Indices, $$f$$ is a homomorphism from $$\left({\mathbb{Z}, +}\right)$$ into $$\left({G, \circ}\right)$$.

Its range $$f \left({\mathbb{Z}}\right)$$ is therefore a subgroup of $$\left \langle {a} \right \rangle$$ containing $$a$$ by Homomorphism Preserves Subsemigroups.

So $$f \left({\mathbb{Z}}\right) = \left \langle {a} \right \rangle$$ because $$\left \langle {a} \right \rangle$$ is the smallest subgroup of $$G$$ containing $$a$$ by Generator of a Group.

Therefore $$f$$ is an epimorphism from $$\left({\mathbb{Z}, +}\right)$$ onto $$\left({G, \circ}\right)$$.