Prove that union of ascending sequence of fields of sets (in the same set $X$) is also field of sets.
I have a sequence $A_1 \subset A_2 ... \subset A_k \subset ...$ of fields of sets. For each $i$ $\ $ $\emptyset \in A_i$, so $\emptyset$ is in $\bigcup_{i=1}^{\infty}A_i$. I am not sure what to do now. The union of finite number of these elements will be always the biggest element. Also I don't know how to work with complements.
It should be "union" rather than "sum"?
For a set $S\in\displaystyle\bigcup_{i=1}^{\infty}A_{i}$, there exists some $i$ such that $S\in A_{i}$. Since $A_{i}$ is a field, then $X-S\in A_{i}$, and hence $X-S\in\displaystyle\bigcup_{i=1}^{\infty}A_{i}$.
For a set $S,T\in\displaystyle\bigcup_{i=1}^{\infty}A_{i}$, there are $i,j$ are such $S\in A_{i}$ and $T\in A_{j}$. Without loss of generality, assume that $i<j$, then $A_{i}\subseteq A_{j}$ and hence $S\in A_{j}$. Since $A_{j}$ is a field, then $S\cup T\in A_{j}$, so $S\cup T\in\displaystyle\bigcup_{i=1}^{\infty}A_{i}$.