I realize that this is a quite simple proof. I can reason the answer but cannot prove it.
problem Statement:
Say c is a real number and has the property Q if x <= y whenever x, y in real numbers with cx <= cy. Which real numbers have property Q?
The problem statement is exactly as I have written it. Perhaps it is not a proof and that simply the answer would suffice. My other rationale is that we need to use a certain property that we know is true i.e. (x-y)^2 > 0 for x != y. If you Plato cannot formalize a proof for this then I am going to quit feeling bad about it. My only problem is that I would expect the prof to say "prove it".
Suppose , .
Case 1:
Which is property Q.
Case 2: c=0
Which is trivially property Q.
Case 3:
Which is only property Q if x=y, but Q is for all , so case 3 does not satisfy property Q.
Thus property Q is being non-negative.