Putnam 1979 B6

Problem B-6 For $k=1,2, \ldots, n$ let $z_{k}=x_{k}+i y_{k}$, where the $x_{k}$ and $y_{k}$ are real and $i=\sqrt{-1}$. Let $r$ be the absolute value of the real part of $$\pm \sqrt{z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}}$$

Prove that $r<\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|$.