Proof: The orbit of any element of a group is a subgroup

Following is a proof that, if $G$ is a group and $g\in G$, then $⟨g⟩\le G$. Here $⟨g⟩$ is the orbit of $g$ and is defined as

$$⟨g⟩=\{g^n:n\in \mathbb{Z}\}$$

Since $g\in ⟨g⟩$, then $⟨g⟩$ is nonempty.

Let $a,b\in ⟨g⟩$. Then there exist $x,y\in \mathbb{Z}$ such that $a=g^x$ and $b=g^y$. Since $a{b}^{-1}=g^x{(g^y)}^{-1}=g^x{g}^{-y}=g^{x-y}\in ⟨g⟩$, it follows that $⟨g⟩\le G$.