If $A_n$ are fields (of sets) satisfying $A_n \subset A_{n+1}$ show $\cup_n A_{n}$ is also a field. If $A_n$ are fields satisfying $A_n \subset A_{n+1}$ show $\cup_n A_{n}$ is also afield. Reminder: In order for something to be a field it must satisfy: 1. $\Omega \in \mathcal{A}$ and $\emptyset \in \mathcal{A}$ 2. if $A \in \mathcal{A}$ then $A^c \in \mathcal{A}$ 3. if $A,B \in \mathcal{A}$ then $A \cup B\in \mathcal{A}$ Proof of 1. If $\mathcal{A}_n$ is a field then $\Omega \in \mathcal{A}_n$. Hence, $\Omega \in \cup_n\mathcal{A}_n$. Similary, for the $\emptyset$.

Define $\mathcal{A}:=\bigcup_{n}\mathcal{A}_{n}$

(1) You allready proved that yourself.

(2) If $A\in\mathcal{A}$ then $A\in\mathcal{A}_{n}$ for some $n$ and consequently $A^{c}\in\mathcal{A}_{n}\subseteq\mathcal{A}$.

(3) If $A,B\in\mathcal{A}$ then $A\in\mathcal{A}_{n}$ for some $n$ and $B\in\mathcal{A}_{m}$ for some $m$ .

If $k:=\max\left\\{ n,m\right\\} $ then $A\in\mathcal{A}_{n}\subseteq\mathcal{A}_{k}$ and $B\in\mathcal{A}_{m}\subseteq\mathcal{A}_{k}$ hence $A\cup B\in\mathcal{A}_{k}\subseteq\mathcal{A}$.