DESIGUALDADES

how to prove that (a+b)(b+c)(c+a)>8abc

First we prove the lemma: 



Proof:
       <----because the square of a real number is 
                            never negative

   <----squaring out the left side

 <----adding  to both sides

     <----factoring the left side 

     <----taking non-negative square roots
                            of both sides

Similarly we can prove 



and



just by changing the letters in the first proof.

Therefore, multiplying the left and right sides of the
three inequalities: