Apply Farkas' lemma to the proposed dual multipliers with the additional constraint $b^T y\le c^T x$. If this system is infeasible, its separating vector consists of a nonnegative vector $d$ and scalar $\lambda$ satisfying $Ad\le\lambda b$ and $c^T d>\lambda c^T x$. For $\lambda>0$, the point $d/\lambda$ improves the alleged optimum. For $\lambda=0$, the point $x+d$ improves it. Both are contradictions. The system is therefore feasible, and weak duality gives equality.