Convergence in Normed Dual Space implies Weak-* Convergence

Theorem
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a convergent sequence in $X^\ast$.

Then $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$.

Proof 1
From Convergent Sequence in Normed Vector Space is Weakly Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly.

From Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$.

Proof 2
Let $f$ 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 $x \in X$: