[Math] Unit ball in dual space is weak*separable

Question

Let $X$ be a separable Banach space, $X'$ the dual space of $X$ endowed with the weak$*$-topology. How to prove that the unit ball $B$ in $X'$ is weak*-separable ?

I only know Banach-Alaoglu, which states that $B$ is compact in the weak*-topology, but not sure if it helps.

Answer 1

Use this question to show that

Theorem. If X is separable, then $B$ endowed with the weak*-topology is metrizable.

Since you also know that $B$ is compact, all you need now is contained in that question:

Theorem. If $X$ is a compact metric space, then $X$ is separable.