Set which is both ring and monotone class it is $\sigma$-ring

Question

Let $S$ be both a ring and monotone class. Show that $S$ is a $\sigma$-ring.

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My attempt

I want to show that $S$ contains $\emptyset$ and it is closed under countable unions and set difference.

1. $\emptyset\in S$ because $S$ is a ring
2. Let $A, B \in S$ then $A \setminus B \in S$ because $S$ is a ring
3. Let $A_k$ be an arbitrary family of sets $\in S$. Define family of sets $B_n = \cup_{i=1}^{n} A_{i}$ then $B_1 \subset B_2 .. \subset B_n ...$ Hence $A = \cup_{i=1}^{\infty} B_{i} \in S$ because $S$ is monotone class.

Is this solution acceptable?