Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequalitystates that, for two sets of real numbers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ , the following inequality is always true:

$\displaystyle({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2$

This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.

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Cauchy-Schwarz Inequality