How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$? The constraints are


*

*$0 \le a_1,a_2....a_n,b_1,b_2....b_n \le 1$.

*$|a_i-b_i|< \delta$ for all $1 \le i \le n $


How do I go about proving the following
$$|\prod_{i=1}^n a_i-\prod_{i=1}^n b_i| < n\delta$$
I tried reducing it to two terms where one term is like $(a-\delta)^n$ so that I can get the $n\delta$ term from binomial,but I am stuck. I would really appreciate just hints.

You can prove that:
$$\left|\prod\limits_{i=1}^{n}a_i-\prod\limits_{i=1}^{n}b_i\right|\le \sum\limits_{i=1}^{n} |a_i-b_i|.$$
The proof goes by induction. For $n=1$ it's obvious. Let's assume that it's true for some $n.$ We will prove this for $n+1.$ Let $A_n=\prod\limits_{i=1}^{n}a_i$, $B_n=\prod\limits_{i=1}^{n}b_i$. We can see that:
$$A_{n+1}-B_{n+1}=(a_{n+1}-b_{n+1})A_n+b_{n+1}(A_n-B_n).$$
The rest goes from the above identity, triangle inequality and from assumption about $a_i,b_i.$
