Cyclic subgroup generated by $a$ is the smallest subgroup that contains $a$ We know that for any element $a\in G$, $\langle a\rangle$ generates a cyclic subgroup $H$ of $G$. Theorem 5.17 from Fraleigh A first course in abstract algebra says that every subgroup containing $a$ contains $H$. I honestly do not understand the proof that he provides. Could someone explain it with extreme detail?

Let $L$ be a subgroup of $G$ such that $a\in L$. Since $L$ is a group and $a\in L$, then $e\in L$ and all the powers of $a$ and $a^{-1}$ belong to $L$. But, since$$\langle a\rangle=\{e\}\cup\left\{a^n\,\middle|\,n\in\mathbb{N}\right\}\cup\left\{(a^{-1})^n\,\middle|\,n\in\mathbb{N}\right\},$$this means that $\langle a\rangle\subset L$.
