$C$ closed under finite intersections then $\sigma(C) \subset D$ Let $\Omega$ be a set and $C, D$ classes of subsets of $\Omega$ satisfying $C \subset D$. $C$ is closed under finite intersections and $D$ contains $\Omega$ and is closed under proper differences and non-decreasing limits.
Prove that $\sigma(C) \subset D$.
 A: I am not completely sure the meaning of proper differences I assumed is necessarily what you mean, but for all sensible meanings I can thing related to differences of sets the proof below should work with minor changes to the third-to-last paragraph.

Since $\Omega\in D$ and $D$ is closed by differences then $D$ is closed by complements. 
Therefore, it is enough to prove that if $A_1,A_2,...\in C$ then $\bigcup_{n=1}^{N}A_n\in D$. Then taking non-decreasing limit we know that $\bigcup_{n=1}^{\infty}A_n\in D$
To produce induction on $N$ it is enough to check the case $N=2$.
Call $B_1=A_1\setminus A_2$, and $B_2=A_2\setminus A_1$, which are both in $D$ due to either being proper differences, or trivially when the sets $A_1,A_2$ are disjoint, or using the closedness by intersection when they are one inside another.
Call $C=A_1\cap A_2$, which is in $D$ due to closedness by finite intersections.
Finally, $A_1\cup A_2=B_1\cup C\cup B_2=(B_1^c\cap C^c\cap B_2^c)$ is in $D$ due to closedness by complements and finite intersections.
