Question

Ajutor, vă rog, dau coroniță

Answer 1

$a_i \in \{-1,0,1\} \Rightarrow |a_i| \in \{0, 1\} \Rightarrow |a_i| \leq 1 \Big|\cdot |x_i| \Rightarrow |a_i|\cdot |x_i| \leq |x_i| \Rightarrow$

$\displaystyle \Rightarrow |a_ix_i| \leq |x_i| \Rightarrow \sum\limits_{i=1}^{n}|a_ix_i| \leq\sum\limits_{i=1}^{n}|x_i|$

$(*)\,\,|x+y| \leq |x|+|y| \,\,\,\text{(inegalitatea triunghiului)}$

$\displaystyle \overset{(*)}{\Rightarrow}\left|\sum\limits_{i=1}^{n}a_ix_i\right| \leq\sum\limits_{i=1}^{n}|a_ix_{i}|$

$\displaystyle \text{Dar } \sum\limits_{i=1}^{n}|a_ix_i| \leq\sum\limits_{i=1}^{n}|x_i| \Rightarrow \left|\sum\limits_{i=1}^{n}a_ix_i\right| \leq\sum\limits_{i=1}^{n}|a_ix_{i}| \leq\sum\limits_{i=1}^{n}|x_i| \Rightarrow$                              

$\displaystyle \Rightarrow \left|\sum\limits_{i=1}^{n}a_ix_i\right| \leq\sum\limits_{i=1}^{n}|x_{i}| \Leftrightarrow |a_1x_1+a_2x_2+...+a_nx_n| \leq |x_1|+|x_2|+...+|x_n|$