Q:

A problem concerning countable intersection

$\forall i\in\mathbb{N}^+,A_i\subset \mathbb{R}$,and $A_i=[i,\infty)$.Then what does 
$$\bigcap_{i=1}^{\infty}A_i$$
mean?

A:

It means what intersection always means: the set of things that belong to every $A_i$, $i=1,2,3,\dots$ . Suppose that $x\in\Bbb R$ and $n\in\Bbb N^+$. Then $x\in A_n$ if and only if $x\ge n$. Thus, $x\in\bigcap_{n=1}^\infty A_n$ if and only if $x\in A_n$ for all $n\in\Bbb N^+$, if and only if $x\ge n$ for all $n\in\Bbb N^+$. But there is no real number $x$ such that $x\ge n$ for all $n\in\Bbb N^+$, so $\bigcap_{n=1}^\infty A_n=\varnothing$.

