Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent/Proof 2

Proof
Let $J$ be the evaluation linear transformation on $X$.

By Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual:
 * $J : X \to X^{\ast \ast}$

Thus, for each $x \in X$: