Supporting hyperplane theorem

Suppose that $C$ is a closed convex set in $\mathbb {R} ^{n}$ and that $y$ is a vector in $\mathbb {R} ^{n}$ that is not in $C$. Then there is a nonzero vector ${\underline {a}}$ that exists in $\mathbb {R} ^{n}$ and a real number $\alpha$ such that ${\underline {a}}\cdot x\leq \alpha \leq {\underline {a}}\cdot y\quad \forall \,x\in C$