Quotient Group of Cyclic Group/Proof 2

Proof
Let $H$ be a subgroup of the cyclic group $G = \left \langle {g} \right \rangle$.

Then by Homomorphism of Powers for Integers:
 * $\forall n \in \Z: q_H \left({g^n}\right) = \left({q_H \left({g}\right)}\right)^n = \left({g H}\right)^n$

As $G = \left\{{g^n: n \in \Z}\right\}$, we conclude that:
 * $G / H = q_H \left({G}\right) = \left\{{\left({g H}\right)^n: n \in \Z}\right\}$

Thus, by Epimorphism from Integers to Cyclic Group, $g H$ generates $G / H$.