Let $X$ be the dual space of a normed space $X$ and let $X^*$ be separable. Prove that $S^* = \left\{F \in X^* : ||F|| = 1\right\}$ is separable.
My solution : suppose $Y$ is a dense countable set in $X^*$, it is enough to prove that $Y \cap S^*$ is dense in $S^*$ . This is true because $\overline{Y \cap S^*} = \overline{Y} \cap S^* = X^* \cap S^* = S^*$.
Is it that simple or am I missing something?
If this is correct, which I doubt, it would mean that every subset of the separable space $X^*$ is separable.