# How does this list of optimal values prove Farkas' lemma?

From Convex Optimization:

Can someone explain how the proof below shows that $$5.87$$ and $$5.88$$ are strong alternatives?

It seems to just list the optimal values of $$5.89$$ and $$5.90$$ depending on the feasibility of $$5.87$$ and $$5.88$$ and claim that they're equal because of this list.

Is there a feasibility relationship between a linear program and it's dual that would suggest that exactly one of $$5.87$$ and $$5.88$$ is feasible?

1. If the dual is feasible, the dual has optimal value 0, and therefore the primal also has optimal value 0 (by strong duality), and therefore you cannot have $$x$$ such that $$c^Tx < 0$$, $$Ax \leq 0$$.
2. If there is an $$x$$ for which $$c^Tx < 0$$ and $$Ax \leq 0$$, then the primal problem is unbounded (fill in $$tx$$ and let $$t\to\infty$$), therefore the dual is infeasible.