 Solution

Fix a primal feasible point $x_0$. For every dual feasible point $y$, nonnegativity and the primal and dual inequalities giveHence the dual objective is bounded below by $c^T{x}_0$ wherever it is feasible, which is weak duality. This is the meaning of “bounded” in this assertion; it does not say that the dual feasible set is bounded, or that it is nonempty. For example, with $A=0$, $b=0$, $c=1$ and a single variable, the primal is feasible and unbounded above, while its dual would require $0\ge 1$ and is infeasible.