Separation Theorems and Supporting Hyperplanes

Show that for the sphere \(S(0, 1)\) in a real normed space \(X\) and any point \(x_0 \in S(0, 1)\) there is a hyperplane \(H_0 \ni x_0\) such that the closed ball \(\widetilde {B}(0, 1)\) lies entirely in one of the two closed half spaces determined by \(H_0\).

Let \(X\) be a normed vector space. Let \(x_0 \in X\) and consider the closed ball \(\widetilde {B}(0,\|x_0\|)\). Use the Hahn-Banach theorem to show that there exists \(f \in X'\) such that \(f(x_0) = 1\) and for any \(x \in \widetilde {B}(0,\|x_0\|)\) we have \(f(x) \le 1\). Do not use Hahn-Banach for convex sets or the separating hyperplane theorem.