Power Set is Closed under Countable Unions

Theorem
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
 * $\forall A_n \in \powerset S: n = 1, 2, \ldots: \ds \bigcup_{n \mathop = 1}^\infty A_n \in \powerset S$

Proof
Let $\sequence {A_i}$ be a countably infinite sequence of sets in $\powerset S$.

Consider an element of the union of all the sets in this $\sequence {A_i}$:
 * $\ds x \in \bigcup_{i \mathop \in \N} A_i$

By definition of union:
 * $\exists i \in \N: x \in A_i$

But $A_i \in \powerset S$ and so by definition $A_i \subseteq S$.

By definition of subset, it follows that $x \in S$.

Hence, again by definition of subset:
 * $\ds \bigcup_{i \mathop \in \N} A_i \subseteq S$

Finally, by definition of power set:
 * $\ds \bigcup_{i \mathop \in \N} A_i \in \powerset S$

Also see

 * Power Set is Closed under Union
 * Power Set is Closed under Intersection
 * Power Set is Closed under Set Difference