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

Since $g \in \langle g \rangle$ , then $\langle g \rangle$ is nonempty.

Let $a,b \in \langle g \rangle$ . Then there exist such that $a=g^x$ and $b=g^y$ . Since $ab^{-1}=g^x(g^y)^{-1}=g^xg^{-y}=g^{x-y} \in \langle g \rangle$ , it follows that $\langle g \rangle \le G$ .