= 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 implies
$$
\boxed{\bigcap_{n\ge1}K_n\ne\varnothing.}
$$
Equivalently, an empty intersection would give an open cover of $K_1$ by their complements, with a finite subcover contradicting nested nonemptiness.
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