Definitions
Let $\mathbf{A}\in \mathbb{R}^{m\times n}, \mathbf{b}\in \mathbb{R}^{m}$ and $\mathbf{a} \in \mathbb{R}^n\setminus\{ \mathbf{0} \}, b \in \mathbb{R}$.
- The set $\{ \mathbf{x} \in \mathbb{R}^n: \mathbf{A}\mathbf{x}\geq \mathbf{b}\}$ is called a Polyhedron.
- The set $\{ \mathbf{x} \in\mathbb{R}^n:\mathbf{a}'\mathbf{x}=b \}$ is called a Hyperplane.
- The set $\{ \mathbf{x} \in \mathbb{R}^n: \mathbf{a}'\mathbf{x}\geq b \}$ is called a Halfspace.
Also let $\mathrm{conv}(X)$ be the set of all convex combinations of points in $X$. We call $\mathbf{x}\in X$ an extreme point of $X$ if $\mathbf{x}\not\in\mathrm{conv}(X\setminus\{x\})$.
Question
(IMPORTANT) For the rest of the question $X\subset \mathbb{R}^n$ is a finite set of points.
I want to prove that for an extreme point $p$ of $X$ there is a halfspace $H$ such that $p$ belong on the boundary of $H$ (the corresponding hyperplane) and all points of $X$ belong to $H$. However I don't assume any knowledge of Linear Programming from the reader.
Basically I want to prove that if $p$ is an extreme point then $p$ is also a vertex the corresponding polyhedron. However that would require me to prove that $\mathrm{conv}(X)$ forms a polyhedron and then prove my result regarding the extreme point and vertices. This would require me to introduce all the LP basics I'll use in the proof, which I won't be using later.
I think the question is too simple to require all those tools, but all my ideas boil down to using LP principles (I am terrible in geometry). I know the obvious thing is to skip the proof but my text follows a pattern that requires me to prove the specific result. Any ideas for how to prove the result without using such methods?