Trying to understand definition of weak topology on normed vector spaces

Question

Given a normed vector space $X$ and $f\in X^{\ast}$ (=dual space of $X$), we define a seminorm on $X$ by $$p_{f}(x):=|f(x)|$$ for all $x\in X$. The book "A Course in Functional Analysis (second edition)" written by "John B. Conway" defines the weak topology on $X$ using the fact that the family $$\mathscr{P}:=\{p_{f} \ | \ f\in X^{\ast}\}$$ of seminorms is separable, i.e. $$\bigcap_{f\in X^{\ast}}p_{f}^{-1}[\{0\}]=\{0_{X}\},$$ or, equivalently, $$\forall x\in X\setminus\{0_{X}\}, \ \exists x^{\ast}\in X^{\ast}, \ p_{x^{\ast}}(x)>0.$$ However, I don't see why this is true. I heard that it can be shown by the Hahn-Banach theorem, but I have no idea where to start. Any suggestions would be greatly appreciated.