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.

Hint: Your idea is good. Apply Hahn-Banach on $x-y\neq 0$.
