Now it's the turn of measure theory

I don't understand something I've read...

We have a sequence of sigma-algebras \(\displaystyle \mathcal{F}_n\).

And \(\displaystyle \mathcal{F}\) is the smallest sigma-algebra containing every \(\displaystyle \mathcal{F}_n\)

There's a property one wants to prove for any \(\displaystyle A_1,\dots,A_k\) in \(\displaystyle \mathcal F\)

And it is said that

I'm not sure : can we write that \(\displaystyle \mathcal F=\bigcup_{n\geq 1} \mathcal{F}_n\) ? I think an uncountable union of sigma-algebras is a sigma-algebra, but I still have some doubts. Otherwise, I know I can just say that \(\displaystyle \mathcal{F}\) is generated by the union.Assume we can prove the desired identity for all n and for all \(\displaystyle A_1,\dots,A_k\in\mathcal F_n\).

By the monotone class theorem, the identity holds true for any \(\displaystyle A_1\in \mathcal F\) and all n and all \(\displaystyle A_2,\dots,A_k\in\mathcal F_n\) and so on...

Next point, I don't understand how the monotone class theorem acts here... I followed a link in the Wikipedia to transfinite induction. I have an intuition that the explanation has something to do with that, but I'm sorry to say that I didn't understand much in it, or rather how to use it here...

So is anyone able to explain it to me, please ? ^^

Thank you