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Prove or Disprove: For any family of sets $\{A_n\}_{n\in\mathbb N}$

$$\bigcup_{n=1}^\infty\mathcal P \left({A_n}\right)\subseteq \mathcal P \left({\bigcup_{n=1}^\infty A_n}\right)$$

How do I approach proving this? I know how to unpack the definition of powersets ($\mathcal P \left({A}\right) = \{x | x \subseteq A\}$) but I'm not sure what else I can do. I've done powerset proofs before but none involving indexed family of sets.

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    Like one approaches many such problems. show that if the set $w$ is an element of the left-hand side, it is an element of the right-hand side.2012-10-25

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