$\left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \leq \sum_{i=1}^{n} x_{i}$ I was wondering if for $x_i\in\mathbb{R}_{\geq0}$ the inequality
$$
\left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \leq \sum_{i=1}^{n} x_{i}
$$
Holds. If so, is there a name for it?
My attempt
$$
\sum_i^n i = \frac{n(n+1)}{2}
$$
$$
\sum_i^n i^2 = \frac{n(n+1)(2n+1)}{6}
$$
Since the right hand side is $O(n^{3/2})$ and the left hand side is $O(n^2)$, and n is a positive integer, the inequality should hold.

I don't know whether there's a name for it, but note that\begin{align}\left(\sum_{i=1}^nx_i\right)^2&=\sum_{i=1}^nx_i^{\,2}+\overbrace{\sum_{i\ne j}x_ix_j}^{\geqslant0}\\&\geqslant\sum_{i=1}^nx_i^{\,2}\end{align}and that therefore$$\sum_{i=1}^nx_i\geqslant\sqrt{\sum_{i=1}^nx_i^{\,2}}.$$
