Prove by induction that $\bigg \vert\prod_{k=1}^{n} a_k - \prod_{k=1}^{n} b_k \bigg \vert \leq \sum_{k=1}^{n} | a_k - b_k|$.

Question

Let $a_k, b_k \in \mathbb{C}$ with $|a_k| \leq 1$ and $|b_k| \leq 1$ for $k = 1,...,n$.

Conclude for $0 \leq s < t$ that $t^n - s^n \leq nt^{n-1}(t-s)$.

Funnily enough I think I already solved the second part of this exercise even though I don't even know where to start with the first part. I used that $t^n-s^n=(t-s)\sum_{k=0}^{n-1}t^{n-k-1}s^{k}$ to show that \begin{align} t^n-s^n&=(t-s)\sum_{k=0}^{n-1}t^{n-k-1}s^{k}\\&<(t-s)\sum_{k=0}^{n-1}t^{n-k-1}t^{k}\\&=(t-s)\sum_{k=0}^{n-1}t^{n-1}\\&=(t-s)nt^{n-1}\end{align} which is sufficient since for $0 \leq s < t$ we also have $0 \leq s^n < t^n$.

Can anybody help me out with the first part?

Possible duplicate of How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$? – the question is more specific, but the answer is exactly what you are looking for. — Martin R, Dec 11 '17 at 06:29

score 4 · Accepted Answer · answered Dec 10 '17 at 22:16

Let $A_{n} = \prod_{k=1}^{n}a_{k}$ and $B_{n} = \prod_{k=1}^{n}b_{k}$.

The base case $n = 1$ obviously holds true.

Suppose we have $|A_{n} - B_{n}| \leq \sum_{k=1}^{n}|a_{k} - b_{k}|$.

Then we have $|A_{n +1} - B_{n+1}| \leq |a_{n+1} - b_{n+1}||A_{n}| + |A_{n} - B_{n}||b_{n+1}| \leq \sum_{k=1}^{n+1} |a_{k} - b_{k}|$ (I have used the triangle inequality, the hypothesis, and the inductive step]. This completes the induction.

score 3 · Answer 2 · answered Mar 18 '18 at 22:59

A different proof, possibly interesting: The problem said "prove by induction". In some sense any proof of anything for every $n\in\Bbb N$ is based on induction at some point. Here the induction is in the "product rule" for the derivative of the product of $n$ functions:

$\left(\prod_{j=1}^nf_j\right)'=\sum_{k=1}^nf_k'\prod_{j\ne k}f_j$.

Now define $$f(t)=\prod_{j=1}^n(ta_j+(1-t)b_j).$$Since $|a_j|\le 1$ and $|b_j|\le 1$ the product rule shows that $$|f'(t)|\le\sum|a_j-b_j|\quad(0\le t\le 1),$$hence $$|f(1)-f(0)|\le\sum|a_j-b_j|.$$

Prove by induction that $\bigg \vert\prod_{k=1}^{n} a_k - \prod_{k=1}^{n} b_k \bigg \vert \leq \sum_{k=1}^{n} | a_k - b_k|$.

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