= Solution
Suppose no $F_N$ is contained in $U$. Then $K_i=F_i\cap(X\setminus U)$ are nested nonempty closed subsets of the <compact> space. Part (ii) gives a point in their intersection. Such a point lies both in $\bigcap_iF_i$ and outside $U$, contradicting the hypothesis. Thus some $F_N\subseteq U$, and nesting gives
$$
\boxed{F_i\subseteq U\quad\text{for every }i\geq N.}
$$
<Compactness> upgrades containment of the limiting intersection to eventual containment of the whole nested sets.
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