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 $q_i \geq 0$.

  • $\begingroup$ 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

Yes. Let $s= \sup_j|x_j|$. Then

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.