Estimates on $|\prod_{i=1}^n u_i - \prod_{i=1}^n v_i|$, $u_i,v_i\in [0,1]$

Question

I have to verify two estimates in the following setting: $u_i,v_i\in [0,1]$ and $|u_i-v_i|\leq v$ for all $1\leq i\leq n$. Precisely, I should prove that $$|\prod_{i=1}^n u_i - \prod_{i=1}^n v_i|\leq nv$$ and $$|\prod_{i=1}^n u_i - \prod_{i=1}^n v_i|\leq 1-(1-v)^n$$ I have already tried some different approaches, but none of them was successful. So some tips would be nice.

Answer 1

Ok, I think I managed to proof the first inequality with a telescope sum: $$|\prod u_i - \prod v_i|=|\sum_{i=1}^n u_1u_2...u_{i-1}(u_i-v_i)v_{i+1}...v_n|\leq \sum_{i=1}^n |u_1u_2...u_{i-1}(u_i-v_i)v_{i+1}...v_n|\leq \sum_{i=1}^n|u_i-v_i|\leq nv$$

Is this proof correct?

And I still have problems with the second inequality. My attempt is to divide both sides by $v$ as I can then write $\frac{1-(1-v)^n}{1-(1-v)}=\sum_{i=1}^n (1-v)^i$. However, I can't figure out how to get link now so that I can prove it. Am I on the right track with this approach?