Solution
= Solution
Take the incomplete <metric space> $X=(0,1)$ with its usual distance, and
$$
\boxed{A_n=(0,1/(n+1)]\subset X.}
$$
Each is nonempty and closed relative to $X$: it is $X\cap(-\infty,1/(n+1)]$. The sequence is strictly descending and its <diameters> are $1/(n+1)\to0$. Its intersection is empty, since every positive point exceeds $1/(n+1)$ for sufficiently large $n$. The excluded limiting point is zero.