= Solution
Part i makes $K'_\varepsilon$ a closed subset of the compact metrizable space $K$, so it is compact metrizable and therefore separable. Choose the stated dense sequence $(f_m)$ and, using part iii, the sequences $(g_{m,n})$ and $(y_{m,n})$. In constructing the universal weakly null sequence $(z_n)$ from part b, retain a tail of each $y_m$ and write $z_{j(m,n)}=y_{m,n}$ along the retained subsequence, where $j(m,n)\to\infty$.
Each
$$
A_N=\bigcap_{n\geq N}\{f\in K:|f(z_n)|\leq\varepsilon/20\}
$$
is weak-star closed. Since $z_n\rightharpoonup0$, every fixed $f\in K$ belongs to some $A_N$, so $K=\bigcup_NA_N$. The compact Hausdorff space $K$ is a <Baire space>, and the <Baire category theorem> implies that some $A_N$ has nonempty relative weak-star interior.
Suppose $K'_\varepsilon=K$. Choose a nonempty relatively open $O\subseteq A_N$ and then $f_m\in O$ by density. Since $g_{m,n}\xrightarrow{w^*}f_m$, eventually $g_{m,n}\in O$. For a sufficiently late retained index, also $j(m,n)\geq N$, and hence
$$
|g_{m,n}(y_{m,n})|
=|g_{m,n}(z_{j(m,n)})|
\leq\varepsilon/20,
$$
contradicting $|g_{m,n}(y_{m,n})|>\varepsilon/16$. Therefore the <One-step Szlenk derivation for a separable dual> gives $K'_\varepsilon\subsetneq K$ whenever $K$ is nonempty.
Solved by gpt-5.6-sol high.
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