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The coordinate evaluations $e_n:x\mapsto x_n$ belong to the dual unit ball of the l-infinity sequence space and separate its points. Therefore the countable separating family metrizes a weakly compact set result makes any nonempty weakly compact $K\subset {\ell }_{\infty }$ a compact metric space in its weak topology. Choose finite $1/n$-nets and take their countable union $D$; it is weakly dense in $K$.

It follows that $K\subset {\overline{D}}^w\subset {\overline{\mathrm{conv}D}}^w$. By Mazur theorem, the last set is ${\overline{\mathrm{conv}D}}^{\Vert \cdot \Vert }$. Finite convex combinations of members of $D$ with nonnegative rational coefficients summing to one form a countable norm-dense subset of $\mathrm{conv}D$: for each fixed finite list, approximate its coefficients in the simplex by rational coefficients, and use the norm triangle inequality. Thus its norm closure is a separable metric space containing $K$.

Every subset of a separable metric space is separable: balls centred on a countable dense set with positive rational radii form a countable base; intersect this base with the subset and choose one point from each nonempty intersection. Consequently the weakly compact subsets of l-infinity are norm separable conclusion isThe empty case is immediate. The convexification matters: the chosen weakly dense set $D$ itself need not be norm dense in a nonconvex $K$.