Convergence in Normed Dual Space implies Weak-* Convergence/Proof 2

Proof
Let $f \in X^\ast$ be the limit of $\sequence {f_n}_{n \mathop \in \N}$, i.e.:
 * $\norm {f_n - f}_{X^\ast} \stackrel{n \to \infty}{\longrightarrow} 0$

Thus, for each $x \in X$: