- #1

sammycaps

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**Let (X,d) be a metric space.**

Let (A

Let (A

_{n})_{n [itex]\in[/itex] N}be a sequence of closed subsets of X with the property An [itex]\supseteq[/itex] A_{n+1}for all n [itex]\in[/itex] N. Suppose it exists an m [itex]\in[/itex] N such that A_{m}is compact. Prove that [itex]\bigcap[/itex]_{n[itex]\in N[/itex]}A_{n}is not empty.I'm wondering if there is a typo here. Take some metric space. The we can set A

_{m}= ∅, and this is compact and closed, so it satisfies the conditions but the intersection is empty. What am I missing, thanks.