Separating and supporting hyperplanes

Let $M$ be a convex subset of a two-dimensional affine space and $a \notin M^{0}$ (see 50.10). Prove that a line can be drawn through the point $a$ such that the set $M$ lies on one side of it.

Let $M$ be a convex subset of an $n$-dimensional affine space $A$ and $a \notin M^{0}$ (see 50.10). Prove that a hyperplane can be drawn through the point $a$ such that the set $M$ lies on one side of it.

Prove that through any point of a closed convex set not belonging to its open kernel, a supporting hyperplane can be drawn.

Let $M$ and $N$ be disjoint closed convex sets in an affine space $A$, one of which is bounded. Prove that there exists an affine linear function $f$ on $A$ such that $f(x)<0$ for all $x \in M$ and $f(y)>0$ for all $y \in N$.

Let $M$ be a compact convex set in an affine space $A$, $N$ a compact convex set in the vector space $L$ of all affine linear functions on $A$, and suppose that for every point $a \in M$ there is a function $f \in N$ such that $f(a) \geqslant 0$. Prove that there exists a function $f_{0} \in N$ such that $f_{0}(x) \geqslant 0$ for all $x \in M$.