Prove $\{f_n\}$ converge to $X^*$ iff $\{f_n\}$ is uniformly convergent to closure of $B(0,1)$

Question

Let $X$ a normed vectorial space. Let $\{f_n\}$ a sequence in $X^*$

Prove $\{f_n\}$ converge to $X^*$ iff $\{f_n\}$ is uniformly convergent to closure of $B(0,1)$

Note: $X^*$ is the topologic dual

My attempt:
$\implies$ Suppose $\{f_n\}$ converge to $X^*$

Then for $\epsilon>0$ exists $N\in\mathbb{N}$ with $n\geq N$ then $||fn-f||_{\infty}<\epsilon$

Note

$$||fn-f||_{\infty}=\sup_{||x||=1}|f_n(x)-f(x)|<\epsilon\implies|f_n(x)-f(x)|<\epsilon$$

This happen $\forall x\in X$ in particular for all$x\in B(0,1)$

Moreover, $f$ is bounded and $B(0,1)$ is bounded.

Here i'm stuck. Can someone help me?