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Let (xi) N i=1 and (yi) N i=1 be real numbers not all zero. Define a quadratic polynomial p(λ), by p(λ) = PN i=1(xi + λyi) 2 . Prove that p(λ) has either two complex roots or a double real root. Use this fact to prove the Cauchy-Schwarz inequality | X N i=1 xiyi | 6 X N i=1 x 2 i  1 2 X N i=1 y 2 i  1 2 . Moreover, prove that equality holds if and only if there exists λ0 ∈ R such that xi = λ0yi for all i ∈ {1, . . . , N

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