I'm trying to prove the following statement:
Suppose $\{A_i\}_{i \in \mathbb{N}}$ is a countable, non-increasing collection of non-empty sets, i.e. $A_i \neq \emptyset$ and $A_i \supseteq A_{i+1}$ for any $i \in \mathbb{N}$, then $\bigcap\limits_{i \in \mathbb{N}} A_i \neq \emptyset$.
I suspect the statement to be true, but I am unsure about what kinds of arguments i can use to show it.
In the finite case, a simple induction argument shows that for any $k \in \mathbb{N}$, $\bigcap\limits_{i = 0}^{k} A_i \neq \emptyset$ (actually $\bigcap\limits_{i = 0}^{k} A_i = A_k$). Since this happens for just any $k \in \mathbb{N}$, can I use this as an arguments that it must also hold for the countable intersection (I suspect not)? or do I need some other line of reasoning?
Any pointers to proof methods that can be applied or proofs of other similar statements will be helpful.