Cauchy Schwarz inequality

Cauchy Schwarz inequality

Let $(a_1,a_2, \dots, a_n)$ and $(b_1,b_2,\dots,b_n)$ be two sequences of real numbers, then

By mathematical induction

Let proposition P(n)

when n=1

suppose P(k) is true. then

$\begin{aligned} \sqrt{\sum_{i=1}^{k+1} a_i^2 \sum_{i=1}^{k+1} b_i^2} &= \sqrt{\left( \sum_{i=1}^k a_i^2 + a_{k+1}^2 \right) \left( \sum_{i=1}^k b_i^2 + b_{k+1}^2 \right)} \\ &\ge \sqrt{\sum_{i=1}^k a_i^2 \sum_{i=1}^k b_i^2} + \sqrt{{a_{k+1}}^2 {b_{k+1}}^2} \quad \cdots (*) \\ &\ge \sqrt{\left( \sum_{i=1}^k a_i b_i \right)^2} + \sqrt{{a_{k+1}}^2 {b_{k+1}}^2} \quad (\because \text{귀납 가정 적용}) \\ &= \sum_{i=1}^k \vert{}a_i b_i\vert{} + \vert{}a_{k+1} b_{k+1}\vert{} \\ &= \sum_{i=1}^{k+1} \vert{}a_i b_i\vert{} \quad \left( \ge \sum_{i=1}^{k+1} a_i b_i \right) \end{aligned}$

other prove method

refer to this