Power Set is Closed under Countable Unions
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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:
- $\ds \bigcup_{i \mathop \in \N} A_i \in \powerset S$
$\blacksquare$