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I'm trying to prove the following inequality: $|a^n - b^n | \leq n M^{n-1}|a-b|$ with $M=\max\{|a|,|b|\}$.

My first attempt was using mathematical induction, but I got stuck. I also tried using the triangle inequality but I can't reach the desired result.

Can someone help please?

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    $\begingroup$ Hint: $a^n-b^n$ is divisible by $a-b$. $\endgroup$ – Math1000 Dec 6 '19 at 22:02
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    $\begingroup$ The right hand term should probably say $nM^{n-1}|a-b|$, right? $\endgroup$ – Qi Zhu Dec 6 '19 at 22:03
  • $\begingroup$ Yes, editing right now. $\endgroup$ – Jorge Arturo Quiroz Dec 6 '19 at 22:04
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$$|a^n-b^n|=|a-b|\left|\sum_{k=0}^{n-1}a^kb^{n-k-1}\right|\leq |a-b|M^{n-1}n$$

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Using the mean value theorem for the function $f(x) = x^n$ we get that there exists some $\theta \in \langle a,b\rangle$ such that $$|a^n - b^n| = |f(a) - f(b)| = |f'(\theta)||a-b| = n|\theta|^{n-1}|a-b| \le nM^{n-1}|a-b|$$ since $|\theta| \le \max\{|a|, |b|\} = M$.

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You can use the triangle inequality like this: $$ |a^n-b^n|\leq |a^n-a^{n-1}b|+|a^{n-1}b-a^{n-2}b^{2}|+\cdots+|ab^{n-1}-b^n|. $$ Each term on the right can be written as $|a-b||a|^j|b|^k$ where $j+k=n-1$, and hence each term on the right is bounded above by $M^{n-1}|a-b|$. Now add.

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