Let $X$ be a normed space. If $X^*$ is separable, then there exists $(f_n)_{n\geq1}\subset X^*$ such that $\|f_n\|_{X^*}=1$ for all $n$ and $\{f_n:n\geq1\}$ is dense in $\{f\in X^*:\|f\|=1\}$.
In turn, why does this imply that there exists $(x_n)_{n\geq1}\subset X$ such that $\|x_n\|=1$ and $f_n(x_n)\geq\frac{1}{2}$ for all $n$ ?