 Solution

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 $({\bigcap }_n{K}_n)\setminus U$, 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.