Claim: Let $\mathscr{C}$ be a non-empty class of sets. Show that for each set $A$ in the $\sigma-\text{ring}$ generated by $\mathscr{C}$, there exists a sequence ${A}_{i\in\mathbb{N}}$ of elements of $\mathscr{C}$ such that $A\subset\bigcup_\limits{i=1}^{\infty}A_i$.
I understand that $\bigcup_\limits{i=1}^{\infty}A_i\subset\sigma-\text{ring}$. But I have no idea on how to prove the claim.
Question:
How should I prove the claim?
Hint: Denote by $\mathcal{R}(\mathscr{C})$ the $\sigma$-ring generated by $\mathscr{C}$. Show that
$$\mathcal{D} := \left\{A \in \mathcal{R}(\mathscr{C}); \exists (A_i)_{i \in \mathbb{N}} \subseteq \mathscr{C}: A \subseteq \bigcup_i A_i \right\}$$
is a $\sigma$-ring. Use $\mathscr{C} \subseteq \mathcal{D}$ to conclude that $\mathcal{R}(\mathscr{C}) \subseteq \mathcal{D}$.