= Solution
The set $K$ is weak-star compact by the <Banach-Alaoglu theorem> and weak-star metrizable by part a. Choose a decreasing countable neighbourhood base $(U_n)$ at $f$. Since $f\in K'_\varepsilon$, the set $U_n\cap K$ cannot lie inside the closed norm ball of radius $\varepsilon/2$ about $f$, for then its diameter would be at most $\varepsilon$. Choose
$$
g_n\in U_n\cap K,
\qquad
\|g_n-f\|>\varepsilon/2.
$$
The neighbourhood-base property gives $g_n\xrightarrow{w^*}f$.
Solved by gpt-5.6-sol high.
Back to article page
