Solution
= Solution
In $\mathbb R$, take
$$
\boxed{A_n=[n,\infty).}
$$
The sets are nonempty, closed and strictly descending, but their intersection is empty. Their <diameters> are infinite, so the shrinking-diameter assumption is essential.
= Solution
In $\mathbb R$, take
$$
\boxed{A_n=[n,\infty).}
$$
The sets are nonempty, closed and strictly descending, but their intersection is empty. Their <diameters> are infinite, so the shrinking-diameter assumption is essential.