$\mathbb{N} ⊇ A_1 \supset A_2 \supset A_3 \supset \cdots$ but $\bigcap_{n=1}^∞ A_n$ is infinite?

Question

What is an example of an infinite intersection of infinite sets is infinite?

I know that the intersection of infinite sets does not need to be infinite. However, I am seeking for an explicit example where it is infinite

I was thinking A n = { [n,infinity) }

Answer 1

Trivially, you could intersect an infinite set with itself. For instance, consider the set of real numbers $\mathbb{R}$. Then $$\bigcap_{i=1}^{\infty} \mathbb{R} = \mathbb{R}.$$ If you want something nontrivial, consider $$\bigcap_{i=1}^{\infty} (-\frac{1}{i}-1,\frac{1}{i}+1) = (-1,1).$$ In general, say you have an infinite collection of sets $A=\{A_1,A_2,...\}$ such that $A_1 \subseteq A_2 \subseteq ...$ Then $$\bigcap_{i=1}^{\infty} A_i = A_1.$$ Update: With the condition $A_n \subset \mathbb{N}$ (proper subset), we can use prime numbers. Let $A_n$ be defined numbers divisible by say $2^n$ or divisible by 3. That is $$A_n := \{x \in \mathbb{N} : 2^n \mid x \text{ or } 3 \mid x\} = \{x \in \mathbb{N} : 2^n \mid x \} \cup \{x \in \mathbb{N} : 3 \mid x \}.$$ Then (without proof) $$... \subset A_3\subset A_2 \subset A_1$$ and $$\bigcap_{i=1}^{\infty}A_i = \{x \in \mathbb{N} : 3 \mid x \}$$ which is infinite.

Answer 2

Let $(B_n)$ be a sequence of sets with empty intersection and $B_{n+1} \subset B_n$, say $B_n =\{n,n+1,...\}$. Now take $A_n =B_n \cup E$. Then intersection of $A_n$ 's is $E$. You can take $E$ to be any set, in particular a finite set or an infinite set.

Answer 3

Let $x_k = 2k$ and define $A_n = \{ \mathbb{N} \setminus \{x_k\}_{k=1}^n \}$.

$$\bigcap_{n=1}^\infty A_n = \text{set of odd natural integers}$$

Answer 4

If $A_n$ is $\mathcal{N}$ without the first $n$ prime numbers, then the $A_n$ are nested subsets with an infinite intersection, the non-prime integers.