# Cauchy-Schwarz Inequality

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 .

This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.