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My intuition tells me that the following equation is true but I can't prove it:

$$|\sum_{i=1}^N q_i x_i|\leq \sup_i|x_i|$$

where $x \in \mathbb{R}$, $|x|$ is x's absolute value, $\sum_{i=1}^N q_i=1$ and $q_i \geq 0$.

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  • $\begingroup$ Can you show us what you have done so far? $\endgroup$ – Kavi Rama Murthy Sep 20 '18 at 10:27
  • $\begingroup$ [Kavi Rama Murthy] It's really a big problem in my research paper. So, I need to program it, since, $q_i$ are unknown, so, I need to prove this for use it. $\endgroup$ – zazar Sep 20 '18 at 10:36
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Yes. Let $s= \sup_j|x_j|$. Then

$|\sum_{i=1}^N q_i x_i| \le \sum_{i=1}^N q_i |x_i| \le \sum_{i=1}^N q_i s=s\sum_{i=1}^N q_i =s$.

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