# Linear Program feasibility

Let $A$ be an $m \times n$ matrix, $b \in \mathbb{R}^n$, and consider the linear program $$\max\{ 0^Tx: Ax = b, x \ge 0\},$$ and its dual $$\operatorname{min}\{y^Tb : y^TA \ge 0 \}.$$ Here $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$. I am trying to show that the first one is feasible if and only if the second one is bounded. I think I have the $(\implies)$ direction, since if there is some feasible $x$, then $y^TA x= y^Tb \ge 0$ for all $y$ such that $y^T A \ge 0$, so the bottom set is bounded below by 0. However I am having trouble with the converse. Any help would be appreciated.

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