Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $ Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2}  $. Well i choose $u=(1,\ldots,1)$ and $v=(a_1,\ldots,a_N)$ whit $a_i$ positive and. Apply $u \circ v \leq |u||v|$. With this now i want to prove that $\sum_{i,j}^{N}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_j}\leq N |\nabla u||\nabla v|$. I know that I am very close , but if somebody can help me, i will aprecciate..thank you

Recall the Cauchy-Schwarz inequality for sums:
$$\left( \sum_1^N a_n b_n \right)^2 \le \left( \sum_1^N a_n^2 \right) \left( \sum_1^N b_n^2 \right)$$
Let $b_n = 1$. Then this simplification results:
$$\left( \sum_1^N a_n \right)^2 \le \left( \sum_1^N a_n^2 \right) N$$
Taking the square root of each side gives the desired result.
