Subsystem of infeasible system of linear inequalities Suppose that we have a system of linear inequalities described by
$$Ax \leq b$$ where $A$ is $m \times n$ matrix.
I want to show that if the system is infeasible then it has a subsystem of at most $\text{rank}(A)+1$ inequalities such that the subsystem is also infeasible.
My is initial thought was to use Farkas Lemma on the original system and modify the certificate $y$ to obtain $y'$ as a certificate for a subsystem of at most $\text{rank}(A)+1$ inequalities, but I am stuck from there, maybe my initial thought it wrong?

HINT:
A system of inequalities $Ax \le b$ is incompatible if and only if the inequality $0 \le -1$ is a positive linear combination the inequalities in $Ax \le b$, that is, if and only if the $n+1$ row $(0,0,\ldots, 0, -1)$ is a positive linear combination of the rows of the augmented matrix $(A,b)$. Note that the rank of the matrix $(A,b)$ is $\le \operatorname{rank} A + 1$.
To conclude, you have to use the following: if a vector $v$ is a positive combination of $m$ vectors $v_1$, $\ldots$, $v_m$, then it is a positive combination of a linearly independent subset of the $v_i$'s.
