Suppose no such $n_0$ exists. Nestedness then makes every $K_n\setminus U$ nonempty. These are nested <compact> sets: $X\setminus U$ is closed and $K_n$ is <compact>. By (i) their intersection is nonempty. But that intersection is $(\bigcap_nK_n)\setminus U$, contrary to the hypothesis. Thus $K_{n_0}\subseteq U$ for some $n_0$, and nestedness gives $\boxed{K_n\subseteq U\text{ for all }n\ge n_0}$ immediately.

Solution

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

 Solution (source code)