Your proof is fine. Here's a more general result. Theorem: For $k\in \Bbb Z,$ we have $$k\Bbb Z\le \Bbb Z.$$ Proof: Let $k\in \Bbb Z$. We use the one-step subgroup test; that is, we show that $\emptyset\neq k\Bbb Z\subseteq \Bbb Z$ and for any $m,n\in k\Bbb Z$, written additively, $m-n\in ...