Is it true to write $|\sum_{i=1}^N q_i x_i|\leq \sup_i|x_i|$?

Question

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 $q_i \geq 0$.

Answer 1

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

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