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 don’t 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: .

--------------------------------------(2)

From (1) and (2) we get:

+

++=+++

= +==

Solution B for Problem 10:

By Cauchy-Schwarz inequality:

Let , , we get .

That is,