= One-step Szlenk derivation for a separable dual
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Suppose $X^*$ is norm separable and $K$ is a nonempty weak-star closed subset of its unit ball. Then the <Szlenk derivation> $K'_\epsilon$ is a proper subset of $K$ for every $\epsilon>0$.
By Banach-Alaoglu, $K$ is compact Hausdorff and hence a Baire space. A countable norm-dense family in $X^*$ gives a cover of $K$ by weak-star closed norm balls of radius $\epsilon/4$. One of these closed sets has nonempty relative weak-star interior. That interior is a neighborhood of some point and has norm diameter at most $\epsilon/2$, so the point is removed by the derivation. This direct argument needs neither a universal weakly null sequence nor an assumed norm-to-weak-star equivalence.
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