I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia:
Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true:
- There exists an $x \in \Bbb R^n$ such that $Ax = b$ and $x \ge 0$.
- There exists a $y \in \Bbb R^m$ such that $A^Ty \ge 0$ and $b^Ty < 0$.
The first direction is quite easy. I assume that there is a vector $y$ and I found a contradiction. To the other direction I have used the fourier-motzkin elimination to reduce the number of variables. I assume that $Ax \le b$ and I do one step from the algorithm. I create a new system $A'x' \le b'$. I know that there exist a non-negative matrix $M$ that is a linear combination of the new system to the original. I have followed the direction of repeating the algorithm $n$ times to eliminate all the variables and create the system: $0 \le b''$.
Now in order this system to be infeasible it must be that: $b''<0$. So I can assume that exist vector $y''$ such that $y''A''=0$ and also $y''b''<0$ because $b'<0$ and $y\ge 0$. Now I can prove that also there is a vector $y$ for the original system. But the repetition of $n$ steps seems to me a bit arbitrary. If I just do one step and create the system $A'x'\le b'$ how I can use it is infeasible?