proof that all cyclic groups are abelian

The following is a proof that all cyclic groups are abelian.

Proof.

Let $G$ be a cyclic group and $g$ be a generator of $G$. Let $a,b\in G$. Then there exist $x,y\in \mathbb{Z}$ such that $a=g^x$ and $b=g^y$. Since $ab=g^x{g}^y=g^{x+y}=g^{y+x}=g^y{g}^x=ba$, it follows that $G$ is abelian.∎