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See the following. Original file is here. What does it mean by "standard compactness argument"?

It seems to say if linear systems $A_1x \ge 0, ...,A_Nx \ge 0, f^Tx = 1$ has no solution, then exists one row $a_p$ from each $A_p,p=1,...,N$ s.t. $a_1x \ge 0, ...,a_Nx \ge 0, f^Tx = 1$ has no solution if the solutions of each of $A_px \ge 0, p=1,...,N$ are all contained in a compact set $Q$.

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Related setup:

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