Statement

Proof

If we have a linear programme given by $A$, $b$, and $c$. From the weak duality theorem we have

$$ c^Tx \leq b^Ty.$$

for all feasible $x$ in the original linear programme and $y$ in the dual linear programme. Therefore if the original linear programme is unbounded then no such $y$ can exist making the dual linear programme infeasible.