Let and let for . Prove
Proof. First,
for all . We then apply the Cauchy-Schwarz inequality to the numbers and . So, Cauchy-Schwarz gives us,
Let and let for . Prove
for all . We then apply the Cauchy-Schwarz inequality to the numbers and . So, Cauchy-Schwarz gives us,
You can also prove it this way:
If and are the arithemetic and geometric means of the sequence , then