Solution
= Solution
If $f\notin K'_\varepsilon$, some relative weak-star neighbourhood $U$ of $f$ in $K$ has $\operatorname{diam}U\leq\varepsilon$. Every $g\in U$ has that same $U$ as a neighbourhood of norm diameter at most $\varepsilon$, so $g\notin K'_\varepsilon$. Hence $K\setminus K'_\varepsilon$ is relatively weak-star open and the <Szlenk derivation> $K'_\varepsilon$ is weak-star closed in $K$.
Solved by gpt-5.6-sol high.