**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, _{}