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 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.