I am looking to prove Dummit and Foote Chapter $1$ problem $27$. The other proofs I have seen are longer so I feel like there is something I'm missing or my proofs aren't as clear as they should be. This is what I have:
Associativity of H is clear since $H \subseteq G$. Closure of H is satisfied since $x^nx^m = x^{n+m} \in H$ as $n+m \in \mathbb{Z}$. Since G is a group and each element $x^n$ is in $G$, $x^n$ has the inverse $x^{-n}$. Note that $x^nx^{-n} = x^{-n}x^n = e$ is in H and so H has the identity element of G. Hence H is a subgroup of G.