If $A_1$ and $A_2$ are two collection of subsets in $\Omega$ (Sample Space), I need to prove that $\sigma(A_1) \subseteq \sigma(A_2).$ I understand that there exist minimal unique $\sigma$-algebras generated by $A_1$ & $A_2$ respectively. However, I am not sure what needs to be demonstrated mathematically, in order to prove the subset status.
I tried to construct an example for this. Let A1={1,2} , A2={1,2,3} , Ω={1,2,3,4}
Then,
σ(A1)={∅,Ω,{1,2},{3,4}}
σ(A2)={∅,Ω,{1,2,3},{4}}
How can I proceed beyond this. I am confused as how to interpret the subsets as opposed to elements.
Appreciate your comments. Thank you.