Sigma-Ring is Closed under Countable Intersections

Theorem
Let $\RR$ be a $\sigma$-ring.

Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a sequence of sets in $\RR$.

Then:
 * $\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$

Proof
From De Morgan's laws: Difference with Intersection:


 * $\ds \bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} = A_1 \setminus \paren {\bigcap_{n \mathop = 2}^\infty A_n}$

From Set Difference with Set Difference:

Combining the previous equalities, it follows that:
 * $\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$