Problem 10


Prove: for any positive integer n.

(Hint: use mathematical induction.)


[Problem submitted by Steve Lee, LACC Professor of Mathematics.]



Solution A for Problem 10:

Prove the inequality by mathematical induction, if you donít know Chauchy-Schwarz inequality.

The inequality is obviously true for n=1.

Assume the inequality is true for n=p, ---------(1)

We will prove the inequality for n=p+1: .


From (1) and (2) we get:



= +==


Solution B for Problem 10:

By Cauchy-Schwarz inequality:

Let, , we get .

That is,