## Theorem: Cauchy-Schwarz Inequality

**Let and be sequences of real numbers.** *Then we have*

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*Proof *: Consider the following quadratic function :

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Note at this point that for all .

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That is is a or `‘ positive quadratic so has one or no roots. That means the roots are real and repeated or complex so that we have where :

Now take square roots (remembering .)

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