# Is this inequality true, and does it have a name?

$$\sqrt{\left(\sum_{i=1}^n a_i^2\right)} \leq \sum_{i=1}^n \sqrt{a_i^2}$$

If my proof is not incorrect, then it is true for $n=2$, but before proving the general case I'd like to know whether it is true and if it does have a name.

$$\sum_{i=1}^{n}|a_i|^2 \leq \left(\sum_{i=1}^{n}|a_i|\right)^2$$ does not have a name, since it is a really trivial inequality: $$\left(\sum_{i=1}^{n}|a_i|\right)^2 = \sum_{i=1}^{n}|a_i|^2 + \sum_{\substack{i,j=1\\i\neq j}}^{n}|a_i||a_j|.$$