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