The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers and , the following inequality is always true:
Equality holds if and only if .
There are many ways to prove this; one of the more well-known is to consider the equation . Expanding, we find the equation to be of the form , where , , and . By the Trivial Inequality, we know that the left-hand-side of the original equation is always at least 0, so either both roots are Complex Numbers, or there is a double root at . Either way, the discriminant of the equation is nonpositive. Taking the discriminant, and substituting the above values of A, B, and C leaves us with the Cauchy-Schwarz Inequality, , or, in the more compact sigma notation,
Note that this also gives us the equality case; equality holds if and only if the discriminant is equal to 0, which is true if and only if the equation has 0 as a double root, which is true if and only if .