Absolutely convex hull of a norming set contained unit ball of dual $X^*$ of a banach space $X$ is $w^*-dense$ in unit ball of dual $X^*$.

Question

Let $X$ be a separable Banach space and $X^*$ denotes it's dual. Let $B \subset B_{X^*}$ be a norming set.

where $B_{X^*}$ denotes unit ball centered at origin of dual $X^*$

Then prove that absolutely convex hull of $B$, i.e. $aco(B)$ is $w^*-dense$ in $B_{X^*}$.

I'm studying a research paper where the author stated that it follows from Hahn banach separation theorem. page 550 first line of proof of lemma 3.3 https://drive.google.com/file/d/1JoqBgBQLtaIDxqtTDsulZBsaUjlkF2K6/view?usp=sharing

I don't know how to prove it and is separability of $X$ is necessary here?