I have given the following theorem : Let $(\phi_n)$ is a sequence on $S_{X^*}$ (unit sphere of topological dual of the Banach space $X$). If $(f_n)$ is weak* convergent to zero then $f_n$ has a basic subsequence i.e. $\exists (f_{n_k})\subset(f_n)$ such that $\{f_{n_k}:k\in\mathbb N\}$ is a Schauder basis of $\overline{span\{f_{n_k}:k\in\mathbb N\}}$.
I am trying to prove that the theorem below is a direct consequence of the given theorem.
Theorem : Let $(x_n)$ is a sequence on $S_{X}$. If $(x_n)$ is weak convergent to zero then $x_n$ has a basic subsequence.
I thought using canonical mapping that is $J:X\to(X^*)^*$ where $(J_x)(f)=f(x), \quad \forall f\in X^*, \forall x\in X$. By using this we have $X$ is homeomorphic to $J(X)\subseteq(X^*)^*$.
I am not sure this way is proper or not. If it is true how can I proceed from there?
Thanks in advance for any help.