The following is a proof of the Farkas Lemma that is creating me quite some problems.
[I presented the all proof simply to point out the notation used by the author.]
My problem is with the last part of the proof (i.e. the last png file with the red frame), namely the part in which the author shows that the procedure terminates.
[Below the proof there are my problems and my thoughts over them.]
[I will highlight the problems with a number.]
First of all, trying to get the structure behind this last part, it seems to me that the author proceeds by contradiction. In particular, step 3 can be translated in the following way: $$\forall a^j \in A (y a^j \geq 0 \Longrightarrow STOP)$$ By assuming it and procedeeing by contradiction we have that exists a $a^j \in A$ such that $y a^j \geq 0$, and that $\forall k,l (k<l \Longrightarrow D^k = D^l)$.
Now, the idea looks to me that $D^p$ and $D^q$ differs at least by $a^s$. Honestly, I am not completely sure on the idea of inserting $a^s$ back after the iteration $q$.
1) I mean, I see why it has to be the case, but is it enough to proceed with it?
2) Slightly changing the focus, but basically staying on the same issue, why do we write $D^p \cap \{ a^{s+1}, \dots, a^n \} = D^q \cap \{ a^{s+1}, \dots, a^n \}$? Is it partly for the reason I mentioned before, namely that $D^p$ and $D^q$ differs at least by $a^s$, and that, following from this remark, this is the most compact notation to highlight what the sets have in common?
Frankly, after this point, things start to be so fuzzy that I even have problems to actually come up with precise questions over my doubts. Moreover I am not sure about the notation (e.g. at the end $D^p \cap \{ a^{s+1}, \dots, a^r \} = D^q \cap \{ a^{r+1}, \dots, a^r \}$).
Is there somebody that can help me?
Any feedback will be greatly appreciated (and please point out my mistakes without any mercy!).
Thank you.