The exercise is the following: Let $X$ be a vector space over the real numbers and $p:X \to \mathbb{R}$ a sublinear functional. Show that if $x_0\in X$ then there exists a linear functional $F:X\to \mathbb{R}$ such that $F(x_0)=p(x_0)$ and $F(X) \leq p(x)$ for al $x\in X$.
I know this has Hahn-Banach Theorem written all over it and that the $F(x_0)=p(x_0)$ if $x_0=0$. I just don't find a way to do it for $x_0\neq 0$. Is there any path to do that?
Edit: The Hahn Banach Theorem that I'm using states the following: Let $M_0$ be a subspace of a Vector Space $X$ over $\mathbb{R}$, $p: X \to \mathbb{R}$ a sublinear functional and $f_0$ a linear functional defined only on $M_0$. Then there exists a linear functional $F: X\to \mathbb{R}$ such that:
1)$F(x)=f_0(x)$, for all $x\in M_0$
2) $F(x)\leq p(x)$, for all $x\in X$
Hint: Consider $M_0=\operatorname{Lin}(x_0)$.