[Math] Triangle inequality: How to prove Minkowski inequality

Question

It is not clear for me how the triangle inequality is used here:

And by the triangle inequality (1.3)
$$\left(\sum\limits_{i=1}^n |\xi_i+\eta_i|^2\right)^{1/2} \le
\left(\sum\limits_{i=1}^n |\xi_i|^2\right)^{1/2} + \left(\sum\limits_{i=1}^n |\eta_i|^2\right)^{1/2}.$$

Especially because the absolute value on the left side is squared and we have taken the square root of all the summation on the left side.could anyone clarify this for me?

Edit:

I think I should have corrected my above question "How to prove Minkowski inequality"?

Thanks!

Answer 1

Hint. By squaring both sides we obtain the equivalent inequality $$\sum_{i=1}^n |\xi_i+\eta_i|^2\leq \left(\left(\sum_{i=1}^n |\xi_i|^2\right)^{1/2}+\left(\sum_{i=1}^n |\eta_i|^2\right)^{1/2}\right)^2$$ that is $$\sum_{i=1}^n |\xi_i|^2+\sum_{i=1}^n |\eta_i|^2+2\sum_{i=1}^n |\xi_i||\eta_i|\leq \sum_{i=1}^n |\xi_i|^2+\sum_{i=1}^n |\eta_i|^2+2\sqrt{\sum_{i=1}^n |\xi_i|^2}\sqrt{\sum_{i=1}^n |\eta_i|^2}.$$ Can you take it from here?