Weak* sequentiality

Question

Suppose we are given a Banach space $E$ such that weak* compact subsets of $E^*$ are weak* sequentially compact (for example this happens when $E$ is separable). Does it follow that if $A$ is a subset of $E^*$ and $f\in \overline{A}^{w^*}$ then $f$ is actually a limit of a sequence from $A$? If not, is there a countable set $B\subset A$ such that $f\in \overline{B}^{w^*}$?

Answer 1

Counterexample given by David Mitra is described in this answer. Another type of counterexample is given here or here and more generally here.