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.