I wish to show that an algebra is closed under finite unions.
Can I do it using induction on the number of unions performed over the elements of my algebra say $\mathcal{A}$ instead of induction on the number of elements in $\mathcal{A}$ involved in the union such as the one used in
understanding closed under finite intersection notation
and in
Is strong induction necessary in proving that an algebra is closed under finite union? ?
Is it more appropriate to do induction in the number of unions, instead on the number of elements?