From sequence of sets to sequence decreasing of sets

Question

Let $\{A_n\}_{n\in\mathbb{N}}$ a sequence of sets.

Is it possible starting from $\{A_n\}_{n\in\mathbb{N}}$ to define a decreasing sequence of sets $\{B_n\}_n$?

If it is possible, how is it done?

Thanks!

Answer 1

Your question is unclear, but perhaps that what you are aiming at is$$B_n=\bigcap_{j=1}^nA_j.$$

Answer 2

We define $$\{B_n\}_{n\in\mathbb{N}}$$ as $$ B_n = A_n$$

$$B_{n+1} = B_n \cap A_{n+1}$$

$$B_{n+2} = B_{n+1} \cap A_{n+2}$$ $$B_{n+3} = B_{n+2} \cap A_{n+3}$$

and so forth, $$B_{n+k} = B_{n+k-1} \cap A_{n+k}$$