Proving x^n<y^n with Michael Spivak's Calculus

[Einherjer]

Just started on Michael Spivaks Calculus, going fine so far, but generalizing to n always messes me up.

Homework Statement

prove that If 0≤x<y, then x^n<y^n, n=1,2,3,...

The Attempt at a Solution

I have managed to to prove that x^2<y^2. But I'm i don't quite get how to generalize to n.

First time posting a question here, so tell me if I've done something wrong.

 

[HallsofIvy]

Are you using a proof by induction? Then $x^{n+1}= x(x^n)< x(y^n)< y(y^n)= y^{n+1}$

But I would think that $x^n- y^n= (x- y)(x^{n-1}+ x^{n-2}y+ x^{n-3}y^2+$$\cdot\cdot\cdot+ x^2y^{n-3}+ xy^{n-2}+ y^{n-1})$ would prove it directly.