An algebraic brickwall (real analysis)

[mynameisfunk]

Goal: to show yⁿ=x

This particular part of the proof supposes that yⁿ>x. So we want
an h>0 such that (y-h)ⁿ>x

yⁿ-(y-h)ⁿ<yⁿ-x

yⁿ-(y-h)ⁿ=(y-(y-h))(y^n-1+y^n-2(y-h)+...+(y-h)^n-1)<hny^n-1

this yields h=(yⁿ-x)/(ny^n-1)


my question: how the heck does one derive h from this?

 

[mathman]

You are missing part of the problem statement. What are x and y?

 

[mynameisfunk]

n is an integer, x and y are reals.

Sorry bout that.

 

[epenguin]

Seems to me you are still missing some of the problem statement. Maybe it needs to be of the form for all or for some something and something there exists something such that yⁿ = x ?

With the somethings being integer and reals in some combination.
I think some for arrangements of somethings and the all/some the statement will be false, for others it will be trivial, maybe there is a nontrivial true case?