Sequential Goldstine approximation for a separable dual ($J{x}_n\stackrel{w^{\ast }}{⟶}F\in B_{X^{\ast \ast }}$)

If $X^{\ast }$ is norm separable, choose a dense sequence $f_j$. Goldstine theorem permits $x_n\in B_X$ with $|f_j(x_n)-F(f_j)|<1/n$ for $j\le n$. Uniform norm bounds extend convergence from these test functionals to all of $X^{\ast }$, proving the displayed limit. If $F\notin JX$, no subsequence of $x_n$ converges weakly in $X$. This explicitly witnesses failure of weak sequential compactness in a nonreflexive space with separable dual.