Weak* convergence and weak convergence in X*

Question

Suppose $X$ is a Banach space and $\{f_n\}$ is a sequence in $X^*$ such that

1. $f_n$ converges weak* to $f\in X^*$ (meaning $\lim_{n\to\infty}f_n(x) = f(x)$ for all $x\in X$), and

2. $\lim_{n\to\infty}\alpha(f_n)$ exists for all $\alpha\in X^{**}$.

Is it true that $\lim_{n\to\infty}\alpha(f_n) = \alpha(f)$ ?

I think in the following way: if limit $\lim_{n\to\infty}\alpha(f_n)=v_{\alpha}$, then there is $g\in X^{***}$ s. t. $v_\alpha = g(\alpha)$. Then, if we take $\alpha$ from canonical image of $X$ into $X^{**}$ ($\alpha\in J(X)$), by the unique of weak* limit we obtain that $g = J_1(f)$, where $J_1$ is canonical embedding $X^{*}$ into $X^{***}$.