Let $\left(X,\|\cdot\|\right)$ be a Banach space. I need to show that if $\exists f:X \to K$ ($K$ is either the real or complex numbers) a bounded linear functional s.t $\forall x\in X \setminus \{ 0\} ,\, |f(x)|< \|f\| \cdot \| x \|$, so $X \neq X^{**}$ ($X^{*}$ is the dual space of $X$).
We got a hint: to use Hahn-Banach for a subspace of $X^{*}$. I thought about it and I don't have any good idea how to prove that, I would be glad to get some help.