Must an infinite intersection of infinite sets be infinite?

Question

If $A_2$ is a subset of $A_1$, $A_3$ is a subset of $A_2$, and this goes on infinitely and all contain an infinite number of elements, then is the intersection from $n=1$ to infinity, infinite as well? Prove if no.

Answer 1

The answer to your question is no; infact, the intersection can easily be $\emptyset$. If we define $A_i = (0,1/i)$ for $ i \in \mathbb{N}$, then we have that

$$ A_1 \supset A_2 \supset A_3 \supset \cdots $$

yet

$$ \bigcap_{i=1}^{\infty} A_i = \emptyset $$

despite all the $A_i$ have uncountably many elements. As Peter's comment says, we have that the proposition is false even if we just consider countable sets.

Answer 2

An example already appears here, but maybe this one is for some purposes simpler. Consider the sets $\{n,n+1,n+2,n+3,\ldots\}$. Their intersection is $$ \{1,2,3,4,5,6,\ldots\}\cap\{2,3,4,5,6,\ldots\}\cap\{3,4,5,6,\ldots\}\cap\{4,5,6,\ldots\}\cap\cdots\cdots. $$ This set is empty.

Another example: Consider the set of all intervals $(0,x)$, where $x>0$. Their intersection is empty.

Another example: Suppose one of the infinitely many sets is the set of all even numbers, and another is the set of all odd numbers. What the others are besides those two, we don't need to know in order to see that the intersection of all of them is empty.