Sigma-Ring is Closed under Countable Intersections

Theorem
Let $\mathcal R$ be a $\sigma$-ring.

Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N} \in \mathcal R$ be a sequence of sets in $\mathcal R$.

Then:
 * $\displaystyle \bigcap_{n \mathop = 1}^\infty A_n \in \mathcal R$

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


 * $\displaystyle \bigcup_{n \mathop = 2}^\infty \left({A_1 \setminus A_n}\right) = A_1 \setminus \left({\bigcap_{n \mathop = 2}^\infty A_n}\right)$

From Set Difference with Set Difference:

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