For X a non-empty subset of a group G, we define the subgroup of G generated by X to be
\begin{equation*}
\langle X \rangle = \bigcap_{X \subseteq H \leq G} H.
\end{equation*}
In other words, \(\langle X \rangle\) is the intersection of all the subgroups of G that contain X. Prove that
\begin{equation*}
\langle X \rangle = \{x_1^{e_1} x_2^{e_2} \dots x_n^{e_n} \mid n \geq 1, x_1, x_2, \dots, x_n \in X, e_1, e_2, \dots, e_n \in \{-1, +1\} \}.
\end{equation*}
