for reals x and y with x greater than y, prove that n*x^(n-1)(x-y)>=x^n-y^n for n>0.
The Attempt at a Solution
Let p(n) be that the statement is true for some n.
base case: obviously follows
inductive step: assume p(k) is true for some k.
Look at (n+1)x^(n-1+1)(a-b) = (n*x^n+x^n)(x-y)
= n*x*x^n - n*y*x^n+x*x^n-y*x^n
I'm stuck here. I know there is a way to do it using the property (x^n-y^n) = (x-y)(some polynomial), but we were specifically asked not to use that and to do it by induction.