Prove the identity
$$
A \cup\left(\cap_{n=1}^{\infty} B_{n}\right)=\cap_{n=1}^{\infty}\left(A \cup B_{n}\right)
$$
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If $x$ belongs to the set on the left, there are two possibilities. Either $x \in A$, in which case $x$ belongs to all of the sets $A \cup B_{n}$. and therefore belongs to the set on the right. Alternatively. $x$ belongs to all of the sets $B_{n}$ in which case. it belongs to all of the sets $A \cup B_{n}$. and therefore again belongs to the set on the right.
Conversely. if $x$ belongs to the set on the right. then it belongs to $A \cup B_{n}$ for all
$n$. If $x$ belongs to $A$. then it belongs to the set on the left. Otherwise. $x$ must belong to every set $B_{n}$ and again belongs to the set on the left.
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