Let $\alpha$ be the finite supremum of the feasible objectives. Apply Farkas' lemma to the original system with the extra constraint $c^T x\ge\alpha$. An infeasibility certificate has a nonnegative multiplier $t$ for this extra row. If $t=0$, it contradicts original feasibility. If $t>0$, divide the remaining multipliers by $t$ to obtain a dual feasible point with objective strictly below $\alpha$. Weak duality then contradicts the supremum property. Thus the threshold system is feasible and attains $\alpha$.