Hi everyone. I feel like I'm really close to the answer on this one, but just out of reach :) I hope someone can give me some pointers(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let [tex]A1 \supseteq A2 \supseteq A3 \supseteq \ldots [/tex] be a sequence of compact, nonempty subsets of a metric space [tex](X, d)[/tex]. Show that [tex] \bigcap A_n \neq \emptyset [/tex]. (Hint: Let [tex] U_n = X-A_n [/tex])

3. The attempt at a solution

I tried showing by contradiction.

Suppose [tex] \bigcap A_n = \emptyset [/tex]

Choose an open subcover [tex] U_n = X-A_n [/tex] (that's supposed to be set minus but I don't know how to do \ in tex). Then [tex] \bigcup U_n = (X-A_1) \cup (X - A_2) \ldots = X - (\bigcap A_n) = X [/tex]

but where's the contradiction? So X is not compact, but that goes without saying and we can't infer much from that. What am I overlooking here? Or is this the wrong approach entirely?

Thank you for your assistance.

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# Homework Help: Compact nested sequences

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