Soo, fix $0\le \u03f5\le 1$. Given $\lambda \ge 1,{x}_{i}\ge 0$, I know that $\sum _{i=1}^{n}{x}_{i}\ge \lambda n$. I also know that $\sum _{i=1}^{n}{x}_{i}^{2}\le {\lambda}^{2}n+\u03f5$. I am trying to prove a multiplicative error on each ${x}_{i}$, mainly something along the lines of

$|{x}_{i}-\lambda |\le f(\u03f5)\lambda $

Where $f(\u03f5)$ is some function of $\u03f5$, say $f(\u03f5)=2\u03f5$. Is there any inequality that would bound that distance?

$|{x}_{i}-\lambda |\le f(\u03f5)\lambda $

Where $f(\u03f5)$ is some function of $\u03f5$, say $f(\u03f5)=2\u03f5$. Is there any inequality that would bound that distance?