# How to show a primal program is unbounded by using weak duality?

In weak duality theorem, we assume $x_i$ and $y_i$ are feasible. But how could we show a primal program is unbounded by this theorem?

Suppose we have a primal program: $\max \mathbf c^\top \mathbf x, \mathbf A\mathbf x \leq \mathbf b, \mathbf x \geq 0$ such that: $\mathbf A$ has no rows which are all zero, $\mathbf c$ is not the zero vector, $c_j \geq 0$ for $j=1,\dots,n$ and $a_{ij} \leq 0$ for $i=1,\dots,m$ and $j=1,\dots,n$.

It seems obvious that this LP is always unbounded, but how to apply weak duality here?

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Weak duality for maximization problems asserts that the if $\mathbf{p}$ is a feasible solution to the dual problem, and $\mathbf{x}$ is a feasible solution to the primal problem, then
$$\mathbf{p}^{T}\mathbf{b} \geq \mathbf{c}^{T}\mathbf{x}.$$