Our set is defined as $$A_n :=\left]0,\tfrac{1}{n}\right[$$Where n∈$N$. What is? $$\bigcap_{n+1}^\infty A_n$$
Solution : Let’s examine $A_1,\,A_2,\,A_3$:
$$A_1:=\left]0,1\right[$$ $$A_2:=\left]0,\tfrac{1}{2}\right[$$$$A_3:=\left]0,\tfrac{1}{3}\right[$$
It's clear that $A_{n+1}\subset A_n$. Let's define this multiple intersection through an index set $I$:$$\bigcap_{i\in I}^\infty A_i:=\{x∣\forall i\in I:x\in A_{i+1}\to x\in A_n\}, I\subseteq \Bbb N,$$
where $\Bbb N$ is the set of natural numbers.
Best Answer
Just to expand on @KaviRamaMurthy's comment: for any $x>0$, there exists $n\in\Bbb N$ with $\frac{1}{n}<x$, so $x\notin A_n$. Hence $\bigcap_nA_n=\emptyset$.