# Analysis Problem - Showing statements are true.

Prove the following.

Let $\{A_n \}_{n \in \mathbb{N}}$ and $\{ B_n\}_{n \in \mathbb{N}}$ be sequences of sets with $$A_1 \subset A_2 \subset A_3 \dots \subset A_n \dots$$ $$B_1 \subset B_2 \subset B_3 \dots \subset B_n \dots$$ then $\left( \bigcup_{n=1}^{\infty} A_n\right ) \cap \left ( \bigcup_{n=1}^{\infty} B_n\right ) = \bigcup_{n=1}^{\infty}A_n \cap B_n$

Similarly, prove that $$A_1 \supset A_2 \supset A_3 \dots \supset A_n \dots$$ $$B_1 \supset B_2 \supset B_3 \dots \supset B_n \dots$$ then $\left( \bigcap_{n=1}^{\infty} A_n\right ) \cup \left ( \bigcap_{n=1}^{\infty} B_n\right ) = \bigcap_{n=1}^{\infty}A_n \cup B_n$

Can anyone help with these two exercises? I've been stuck on them for awhile but haven't been able to make any significant progress in my notebook.

• Hint: replace every occurence of $\infty$ by $k \in \mathbb{N}$ and work on that. What can you say about for example $\bigcup_{n=1}^{k} A_n ?$ – Shakespeare Sep 17 '14 at 9:23
• Or even just $(A_1\cup A_2)\cap(B_1\cup B_2)$ – David Peterson Sep 17 '14 at 9:36

If $x\in\left(\bigcup_{n=1}^{\infty}A_{n}\right)\cap\left(\bigcup_{n=1}^{\infty}B_{n}\right)$ then $x\in A_{m}$ and $x\in B_{k}$ for positive integers $m,k$.
Note that $x\in A_{m}\subset A_{n}$ and $x\in B_{k}\subset B_{n}$ if $n$ exceeds $k$ and $m$.