If $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$. Let $X$ be a normed linear space. Prove that if $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$.
Here $X^*$ denotes the dual space of $X$.
I am getting some smell of using Hahn Banach theorem but not able to prove it. Need some hints.

Let $x,y \in X$ such that $x \neq y$.
Assume that $f(x)=f(y) ,\forall f \in X^*\Rightarrow f(x-y)=0$
From the consequences of Hahn-Banach exists $f_0 \in X^*$ such that  $||f_0||=1$ and $f_0(y-x)=||x-y||$.
But $$f_0(x-y)=0 \Rightarrow ||x-y||=0 \Rightarrow x=y$$
 contradicting our hypothesis that $x \neq y$
