= Sequential Goldstine approximation for a separable dual
{title2=$Jx_n\overset{w^*}{\longrightarrow}F\in B_{X^{**}}$}
If $X^*$ 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 $X^*$, 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.
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