Show that http://stuff.daniel15.com/cgi-bin/ma...+b_n%29%5E2%7D for all real values of the variables. Furthermore, give a condition for equality to hold.

:Base case

Let http://stuff.daniel15.com/cgi-bin/mathtex.cgi?n%20=%202

http://stuff.daniel15.com/cgi-bin/ma...+b_2%29%5E2%7D

http://stuff.daniel15.com/cgi-bin/ma...b_1+b_2%29%5E2

http://stuff.daniel15.com/cgi-bin/ma..._2%5E2-b_2%5E2

http://stuff.daniel15.com/cgi-bin/ma...a_1a_2+2b_1b_2

http://stuff.daniel15.com/cgi-bin/ma...+b_1b_2%29%5E2

Which is true because of Cauchy-Schwarz.

:Inductive Hypothesis

Assume the inequality is true for http://stuff.daniel15.com/cgi-bin/mathtex.cgi?n%20=%20k

http://stuff.daniel15.com/cgi-bin/ma...+b_k%29%5E2%7D

is true.

:Proof

Need to prove it's true for http://stuff.daniel15.com/cgi-bin/ma...gi?n%20=%20k+1

http://stuff.daniel15.com/cgi-bin/ma...1%7D%29%5E2%7D

So let's add http://stuff.daniel15.com/cgi-bin/ma...Bk+1%7D%5E2%7D to our inductive hypothesis.

This yields:

http://stuff.daniel15.com/cgi-bin/ma...Bk+1%7D%5E2%7D

now what?