rewrite the lhs as
aa² + bb² + cc² + dd² ≤ 1
then it becomes obvious that we are dealing with the sum of four quadratics, each of which has a maximum of 1/4 when the variable is 1/2. i.e. (1/2)(1/2)²=1/4.
The sum of the 4 numbers, each at most 1/4, is at most 1.

Posted by Jer
on 20130510 12:05:59 