I am stuck at the following exercise:
The primal linear program
\begin{align} (P): \max \quad &c^Tx \\ \text{ s.t.} &Ax \le b,\\ &x \ge 0 \end{align}
has an optimal solution for all $b$ if and only if for its dual holds:
\begin{align} (D): \min \quad &b^Ty \\ \text{ s.t.} &A^Ty \ge c,\\ &y \ge 0 \end{align}
has at least one feasible point $y$ and there exists a $C \in \mathbb{R}$ such that $\lvert\lvert{y}\rvert\rvert_\infty \le C$.
So far I was able to come up with the following:
$\Longrightarrow:$ The fact that there are feasible points for $(D)$ is a direct consequence of the Duality Theorem, but I do not see why there should be such a $C$.
I have no idea for the other direction. Could you give me a hint?