Example from Real analysis and probability

There is the definition of sequence of sets being $\downarrow \varnothing$.

$A_n:=[n,\infty)$. $A_n = [n,\infty), A_{n+1} = [n+1,\infty)$, so $A_{n+1} \supset A_n$. But $\cap_n A_n = A_n$. On the other side $A_n \neq \varnothing$. So for me it seems like $A_n:=[n,\infty)$ is not $\downarrow \varnothing$. Could you please explain.