Estimations in a ordered field? My Problem: I am stuck with a proof strategy on the following:
So i have got an ordered field $ (K,+,*,<)  $ given. I also have 
$x,y\in K$
and $0\le y < x$
I have to proof that, for every n $\in \mathbb{N}, n \ge 2$ there is:
$ny^{n-1} < \frac{x^n-y^n}{x-y} < nx^{n-1}$
What i got so far:
It can be easily proven that
$ny^{n-1} < nx^{n-1}$. It won't help us much, i guess.
We also know that the fraction in the middle is >0 because nominator and denominator will always stay positive (because $0\le y < x$). This might be of some use.
This inequality just doesn't ring a bell. I know i have to estimate the fraction somehow. Also i wonder why the task is to prove it for a specific field, not just for $\mathbb{R}$ (What difference does it make?)
Any light you can shed on this is much appreciated.

Since $\;x-y>0\;$ , we have that
$$ny^{n-1}<\frac{x^n-y^n}{x-y}<nx^{n-1}\iff ny^{n-1}(x-y)<x^n-y^n<nx^{n-1}(x-y)$$
and  for example (left inequality):
$$ny^{n-1}(x-y)<x^n-y^n\iff nxy^{n-1}-ny^n<x^n-y^n\iff y^n(1-n)<x^n-nxy^{n-1}$$
But 
$$x^n-nxy^{n-1}>x^n-nx^n=x^n(1-n)>y^n(1-n)$$
using what you mention that "can be easily proven", and thus we're done.
Try to do the rightmost inequality in the same way.
