Call a topological space $X$ Artinian if every nested sequence of closed sets
$$C_1 \supset C_2 \supset C_3 \supset \cdots$$
is eventually constant.
Prove that if $X$ is Artinian then it is also compact.
I don't have a concrete strategy for attacking this. I am assuming (maybe wrongly so) that appealing to the finite intersection property might be of some help. Is this the case? Should I employ a different strategy?
Thank you very much in advance!