Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / E / 4 / b / i / Solution

Compact subsets of a Hausdorff space are closed. Regard every $K_n$ as a closed subset of the compact set $K_1$. Any finite intersection equals the member with largest index and is therefore nonempty. The finite intersection property for closed subsets of a compact space impliesEquivalently, an empty intersection would give an open cover of $K_1$ by their complements, with a finite subcover contradicting nested nonemptiness.