Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable

Theorem
Let $X$ be a separable normed vector space.

Let $X^\ast$ be the normed dual space of $X$.

Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.

Let $w^\ast$ be the weak-$\ast$ topology on $B^-_{X^\ast}$.

Then $\struct {B^-_{X^\ast}, w^\ast}$ is separable.

Proof
From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.

From Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struct {B^-_{X^\ast}, w^\ast}$ is metrizable.

From Compact Metric Space is Separable, we have that $\struct {B^-_{X^\ast}, w^\ast}$ is separable.