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

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.

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

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