Union of Countable Sets of Sets/Proof 1

Proof
Since $\AA$ is countable, its contents can be arranged in a sequence:
 * $\AA = \set {A_1, A_2, \ldots}$

Let $B \in \BB$.

Consider the sequence of sets:
 * $\sequence {A_1 \cup B, A_2 \cup B, \ldots}$

We may leave out any possible repetitions, and obtain a countable set:
 * $\set {A \cup B: A \in \AA}$

for every $B \in \BB$.

Thus as $B$ varies over all the elements of $\BB$, we obtain the countable family:
 * $\sequence {A \cup B: A \in \AA}_{\paren {B \mathop \in \BB} }$

From Countable Union of Countable Sets is Countable, their union is countable.