Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 2 / a / 2 / Solution

This statement is also true. The preceding compact metrizable ball $B_{X^{\ast }}$ is separable. Each ball $n{B}_{X^{\ast }}$ is therefore weak-star separable, andThe union of countable dense subsets of these balls is a countable weak-star dense subset of $X^{\ast }$.