Lemma 4.7 in Ergodic Theory and Differentiable Dynamics by Racon, makes the statement,
Let $A$ be in the $\sigma$-algebra generated by $\lim_{n\to\infty}\bigvee_{i=0}^n\mathscr{A}_i$. Then for all $\varepsilon>0$, there exists an $N$, such that for all $n \geq N$, the union of atoms $A_n$ of $\mathscr{A_n}$, we have $$ \mu(A_n \Delta A) < \varepsilon $$
The proof there is very small, and it doesn't quite make sense to me, the proof given is
The set of all such $A$ clearly contains $\bigcup_{i\geq 1}\mathscr{A}_i$. This is easily seen to form a $\sigma$-algebra.
This confuses me more than answering the question. How does it contain $\bigcup_{i\geq 1}\mathscr{A}_i$? and how does that form a $\sigma$-algebra? and above all of that, even if it does form a $\sigma$-algebra, how does that prove the statement?
I can't understand how to approach the proof by myself, and the proof here makes absolutely 0 sense to me for now. Can someone provide a proof for this lemma?