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Prove the following inequality

$\ xX + yY \leq \sqrt{ x^2 + y^2}\cdot\sqrt {X^2 + Y^2}$

where $x, y, X$ and $Y$ are real numbers.

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  • 1
    $\begingroup$ Hint: proof by contradiction. $\endgroup$ – Newb Feb 3 '14 at 1:17
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(1) Make square both sides

(2) Use the fact that $$2AB \leq A^2+ B^2$$

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Use Cauchy-Schwarz Inequality for $\mathbb{R}^2$ with the "vectors" $(x,y)$ and $(X,Y)$.

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