If group $G$ is cyclic, then quotient group $G/H$ is cyclic.

Question

Could someone please verify my following proof? I am using (https://math.stackexchange.com/q/1317875)'s idea for the proof, but no one upvoted their answer, so I am not sure whether it is valid.

If group $G$ is cyclic, then quotient group $G/H$ is cyclic.

Let $G=\left \langle g \right \rangle$. Then for all $g'\in G$, there exists an integer $k$ such that $g'=g^{k}$. By definition of $G/H$, $g'H=g^{k}H=(gH)^{k}$ for any $g'H\in G/H$. By definition, $G/H=\left \langle gH \right \rangle$ and so it is cyclic.

Answer 1

Your proof is fine and generalizes to this:

If $\phi: G \to \Gamma$ is a group homomorphism and $G$ is a cyclic, then the image of $\phi$ is cyclic.

We can then take $\Gamma = G/H$ and $\phi$ the canonical projection $g \mapsto gH$, which is surjective.