Quotient Group of Cyclic Group/Proof 2

Proof
Let $H$ be a subgroup of the cyclic group $G = \gen g$.

Then by Homomorphism of Powers for Integers:
 * $\forall n \in \Z: \map {q_H} {g^n} = \paren {\map {q_H} g}^n = \paren {g H}^n$

As $G = \set {g^n: n \in \Z}$, we conclude that:
 * $G / H = q_H \sqbrk G = \set {\paren {g H}^n: n \in \Z}$

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