\(\displaystyle \sum_{i=1}^{n}(x_{i})^{2} \leq \sum_{i=1}^{n} (x_{i})^2 + 2\sum_{i \ne j}^{n} |x_{i}||x_{j}| = (\sum_{i=1}^{n}|x_{i}|)^2\).

And taking the square root it's supposed to grant the result.

Now, I do not see why \(\displaystyle a^2 \geq b^2 \Rightarrow a \geq b\)

Thanks in advance