Prove the following theorem with induction

Question

Prove by induction that if $n$ is a natural number and $x, y, z$ are real numbers such that $|x| \leq z$ and $|y| \leq z$ then $$|x^n −y^n | \leq nz^{n−1}|x−y|.$$

I need to be able to solve this without using any forms of derivatives, though it is similar to the mean value theorem.

Answer 1

Think of $x^n-y^n$ as a polynomial in $x$.

$x=y$ is a solution for this polynomial. Hence you can write $x^n-y^n = (x-y) P(x,y)$ for some polynomial $P$ in $x$ and $y$.

It's not too hard to construct $P$ in fact we have$^1$

$$P(x,y) = x^{n-1}+yx^{n-2}+y^2x^{n-3}+...+y^{n-2}x+y^{n-1}$$

Using the triangle inequality you have

$$|P(x,y)| = |x^{n-1}+yx^{n-2}+y^2x^{n-3}+...+y^{n-2}x+y^{n-1}| \leq |x|^{n-1}+|y||x|^{n-2}+...+|y|^{n-1}$$ each of these terms is smaller than $z^{n-1}$ and there are $n$ terms.

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1. You can construct $P$ by guessing and then fixing, that is, we need to multiply $(x-y)$ with something that gives $x^n-y^n$ so first multiply by $x^{n-1}$ you will get $x^n$ and another irrelevant term $yx^{n-1}$ so we add $yx^{n-2}$ to get read of this term but then we get another term and you keep going. Another way is to write $P(x,y)=\sum_{k,l} a_{k,l} x^ky^l$ and open up the brackets.