Question
If $A_i$ is measurable, prove that $$B=\bigcap_{i=1}^\infty \bigcup_{n=i}^\infty A_n \quad C=\bigcup_{i=1}^\infty \bigcap_{n=i}^\infty A_n$$ is a measurable set.
Attempt
Since a measurable set is defined on a $\sigma$-algebra of measurable sets, then the countable intersection and countable union of measurable sets is also in the $\sigma$-algebra. Since all $A_i$ are measurable sets, it follows that $B$ and $C$ are also measurable sets.
I want to know if this is a valid argument. Am I missing some additional complexity to this question that demands a more involved answer?