Recall the Cauchy-Schwarz inequality,
For arbitrary real numbers and we have
The claim is then that the equality sign holds if and only if there is a real number such that for each .
Proof. () If for all , then equality clearly holds. Assume then that for at least one .
Then, considering the equation and defining,
But, since we know (by assumption), we have which is in (since since for at least one and each term in nonnegative, so the sum is strictly positive).
() Assume there exists such that for each . Then, . So,