While proving this I have proved that Optimal solution cannot lie inside the feasible set and that each supporting hyperplane to a set bounded from below (which is the case as in LPP we can always set x>0) has at least one extreme point.
I have also been give a hint to use Hahn Banach Theorem which states that Every boundary point of a convex set has a supporting hyperplane.
So using these two I now have to prove that any boundary point which is not an extreme point will have an extreme point as the solution (which I assume will be the extreme point lying on the optimal plane (which will also be the Supporting hyperplane or in general every point on this hyperplane will be an optimal solution).
Best Answer
To prove this let x be a boundary point which is not an extreme point but is an optimal solution and P be its optimal hyperplane.
First observe that any optimal hyperplane will be a supporting hyperplane. For this let there be a point y on optimal hyperplane which lies inside the feasible set. Then by definition (c=cost coefficients) cy=cx=optimal value. But this means the optimal value of the Lpp lies inside the Feasible set. Hence contradiction. Thus Optimal Hyperplane=>supporting Hyperplane.
And since each supporting hyperplane has at least one extreme point, P contains an extreme point. Let it be z. since z lies on the plane it is the optimal solution. But this means each point on the line segment x and z is an optimal solution (for this take any general point w between x and z and show cw=cx(=cz=optimal solution)).
Hence proved.
Source: My college teacher.