Can the intersection of a sequence of nested open intervals be nonempty and have finite cardinality?

Question

Clearly you can create a sequence of nested open intervals whos intersection is an interval, but what about an intersection with a single finite element? My main question is whether $\bigcap_{n=1}^\infty I_n $, where $I_n= (b - 1/n , b + 1/n)$, would have $b$ as an element or not? Also why this is or isn't the case?

Answer 1

An infinite intersection of open sets may not be open. Proof that $b \in \cap_{n=1}^\infty I_n$:

$b - \frac{1}{n} < b < b + \frac{1}{n},\forall n \in \mathbb{N}$

$b \in I_n, \forall n \in \mathbb{N}$

$b \in \cap_{n=1}^\infty I_n$ $$\tag*{$\blacksquare$}$$

Can you prove that $b$ is the only element in $\cap_{n=1}^\infty I_n$?