I'm taking my first class in Linear Programming. The book I am reading from is good in that it uses a lot of examples, but bad in that it provides few proofs. I need a proof for the following theorem.
Theorem:
If the constraint set $S$ of a canonical maximization or a canonical minimization linear programming problem is bounded, then the maximum or minimum value of the objective function is attained at an extreme point of $S$.
Glossary of Terms:
Definition 1
The problem
Maximize $f(x_1,x_2,\cdots,x_n)=c_1x_1+c_2x_2+\cdots+c_nx_n$
Subject to $a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\leq b_1$
$a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n \leq b_2$
$\cdots$
$a_{m1}x_1+a_{m2}x_2+ \cdots + a_{mn}x_n \leq b_n$
$x_1,x_2, \dots, x_n \geq 0$
is said to be a canonical maximization linear programming problem. The definition for minimization is analogous.
Definition 2
Let $x= (x_1,x_2,\cdots ,x_n), y=(y_1,y_2,\cdots ,y_n)\in$ R$^n$. Then $tx+(1-t)y$ for $0\leq t\leq 1$ is said to be the line segment between $x$ and $y$ inclusive.
Definition 3
The set of all points $(x_1,x_2, \cdots, x_n)$ satisfying the constraints of the canonical maximization problem is said to be the constraint set
Definition 4
Let $S$ be a subset of R$^n$. $S$ is said to be convex if, whenever $x$=$(x_1,x_2,\cdots,x_n)$,$y$$=(y_1,y_2,\cdots,y_n)\in S$, then
$tx+(1-t)y \in S$ for $0\leq t \leq 1$
Definition 5
A subset $S$ of R$^n$ is said to be bounded if there exists $r\geq 0$ such that every element of $S$ is contained in the closed ball of radius $r$ centered at the origin. A subset of R$^n$ is unbounded if it is not bounded.
Definition 6
The function $f(x_1,x_2,\cdots,x_n)$ is called the objective function of a canonical linear programming problem.
Definition 7
Any element of the constraint set is said to be a feasible point or feasible solution. Any feasible solution which maximizes/minimizes the objective function is said to be an optimal solution.
My Work
I genuinely have no idea how to do this problem. It wasn't assigned as homework, I just want to understand the reasoning behind it because that will help me do better in my course.