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Is there a simple (possible inductive) proof for $|\mathbf{v}| \leq \sqrt{n}(|v_1|+|v_2|+ \cdots +|v_n|)$, $\mathbf{v} \in \mathbf{R}^n$? I've tried Cauchy-Schwarz, it doesn't seem to work.

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up vote 15 down vote accepted

Since $|\mathbf{v}|^2=v_1^2+\ldots+v^2_n\leq (|v_1|+\ldots+|v_n|)(|v_1|+\ldots+|v_n|)$, it is enough to take the square root on both sides of the inequality. Remark: the prefactor $\sqrt{n}$ is superfluous.

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