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?
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 $\displaystyle x \leq y$, $\displaystyle c\in \mathbb{R}$.
$\displaystyle x \leq y$
$\displaystyle 0\leq (y-x)$
Case 1: $\displaystyle c\in \mathbb{R}^+$
$\displaystyle c0\leq c(y-x)\Rightarrow 0\leq cy-cx \Rightarrow cx \leq cy$
Which is property Q.
Case 2: c=0
$\displaystyle 0\cdot 0\leq 0(y-x)\Rightarrow 0\leq 0$
Which is trivially property Q.
Case 3: $\displaystyle c\in \mathbb{R}^-$
$\displaystyle c0\geq c(y-x)\Rightarrow 0\geq cy-cx \Rightarrow cx \geq cy$
Which is only property Q if x=y, but Q is for all $\displaystyle x,y \in \mathbb{R}$, so case 3 does not satisfy property Q.
Thus property Q is being non-negative.