This might be very trivial to show. But I still cannot figure it out. Let $a \in [-1, 1]$ and $b_i, c_i \in \mathbb R$ with $i \in \mathbb N$. Show that $$\sum_i ab_ic_i \leq |\sum_i ab_ic_i| \leq |\sum_i b_ic_i|.$$ How to show the second inequality, please? Thank you!
2026-03-31 10:45:42.1774953942
How to prove the second inequality
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$$\sum_i ab_ic_i \leq |\sum_i ab_ic_i| \leq |\sum_i b_ic_i|.$$
We know that $a \in [-1,1] \Rightarrow |a| \leq 1$
And $a$ does not depend from $i$ so it can go out of the sum.
$$\sum_i ab_ic_i \leq |\sum_i ab_ic_i| =|a\sum_i b_ic_i|\leq |a||\sum_i b_ic_i| \leq 1 \cdot |\sum_i b_ic_i|.$$