One-step Szlenk derivation for a separable dual

Suppose $X^{\ast }$ is norm separable and $K$ is a nonempty weak-star closed subset of its unit ball. Then the Szlenk derivation $K_{ϵ}^{\prime }$ is a proper subset of $K$ for every $ϵ>0$.

By Banach-Alaoglu, $K$ is compact Hausdorff and hence a Baire space. A countable norm-dense family in $X^{\ast }$ gives a cover of $K$ by weak-star closed norm balls of radius $ϵ/4$. 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 $ϵ/2$, 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.