I was just googling around and I came across this problem. Let (X,d) be a metric space. Let (An)n [itex]\in[/itex] N be a sequence of closed subsets of X with the property An [itex]\supseteq[/itex] An+1 for all n [itex]\in[/itex] N. Suppose it exists an m [itex]\in[/itex] N such that Am is compact. Prove that [itex]\bigcap[/itex]n[itex]\in N[/itex]An is not empty. I'm wondering if there is a typo here. Take some metric space. The we can set Am = ∅, and this is compact and closed, so it satisfies the conditions but the intersection is empty. What am I missing, thanks.