Inequality of small two numbers with power

Question

Let $0<x,y<1$ are two real numbers and $n\in\mathbb N$. Is the following inequality true $$x^n-y^n\leq x^{n+1}-y^{n+1}?$$

I split into two cases:

Case1: when $y<x$.

Case2:when $x<y.$

But in both the cases, I can't conclude anything.

Answer 1

If its true for all $x,y$ with $0<x,y<1$, then it must also be true for all $x,y$ with $0 \leq x, y \leq 1$, since both $x^n-y^n$ and $x^{n+1}-y^{n+1}$ are continuous functions.

What happens if you let $y=0?$