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:

$\ds \bigcup_{i \mathop \in \N} A_i \in \powerset S$

$\blacksquare$


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