Cauchy-Schwarz Inequality

Cauchy’s Inequality can actually be proved by purely noticing the following identity.

Lagrange's Identity

${\left({\sum }_{i=1}^n{a}_i{b}_i\right)}^2={\sum }_{i=1}^n{a}_i^2{\sum }_{i=1}^n{b}_i^2-\frac{1}{2}{\sum }_{i=1}^n{\sum }_{j=1}^n{\left(a_i{b}_j-a_j{b}_i\right)}^2.$