The union of powersets is contained in the powerset of union 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.
 A: The first thing to try when you’re faced with proving an inclusion like this is to show that each element of the lefthand side is an element of the righthand side. That is, begin by letting $x$ be an arbitrary element of $\bigcup\limits_{n\in\Bbb N}\wp(A_n)$. Then start applying definitions. In order for $x$ to belong to a union, it must belong to one of the sets whose union is being taken, so if $x\in\bigcup\limits_{n\in\Bbb N}\wp(A_n)$, then $x\in\wp(A_k)$ for some particular $k\in\Bbb N$. Is that enough to ensure that $x\in\wp\left(\bigcup\limits_{n\in\Bbb N}A_n\right)$?
Added: Yes, but there’s still some work to be done. Since $x\in\wp(A_k)$, $x\subseteq A_k$. But $A_k\subseteq\bigcup\limits_{n\in\Bbb N}A_n$ (why?), so $x\subseteq\bigcup\limits_{n\in\Bbb N}A_n$, and hence $x\in\wp\left(\bigcup\limits_{n\in\Bbb N}A_n\right)$. The amount of justification that you’ll need for these steps depends on what you’ve already proved; for instance, you may need to justify the fact that if $x\subseteq y\subseteq z$, then $x\subseteq z$.
