I was just googling around and I came across this problem.(adsbygoogle = window.adsbygoogle || []).push({});

Let (X,d) be a metric space.

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.

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# Homework Help: Sequence of Compact Spaces

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