One-step Szlenk derivation for a separable dual

If $X^{\ast }$ is separable and $K$ is a nonempty weak-star closed subset of $B_{X^{\ast }}$, then $K_{\epsilon }^{\prime }$ is a proper subset of $K$ for every $\epsilon >0$. Metrizability produces sequences witnessing membership in the derivative, while a Baire-category argument applied to a universal weakly null sequence yields the strict inclusion.