Since the inequality we are to prove for the letters
representing unequal positive numbers:


and one of the other parts is



Then we might suppose that the first thing we need to
prove is that for any different positive numbers a and b 



The technique I will use is to assume that it is false,
and then reach a contradiction.  

So for contradiction we will assume that for ,  
 


Multiply both sides by 2



Square both sides:





Subtract 4ab from both sides:



Factor the left side:



This is false since the left side is positive,
since a and b are different.

So we have reached a contradiction and therefore
we have proved that 



Note that the two sides are equal if a=b.

We have also proved by letting a=c and b=d.
that for  



Down below we will need this for positive x,y: 



with equality holding if and only if x=y.

We do this to avoid getting letters confused.


--------------------------------

Next we will prove that for a,b,c,d > 0, all different,



We start with




Multiply both sides by 2, and we have:




Add the two inequalities:



Factor out 2 on the right



Make the left side into the left side of what
we have to prove by multiplying thru hy 1/4



Now we will show that the right side is greater than or equal
to 

We recall from above that for positive x, y,



with equality holding if and only if x=y

We let x = 
and y = 





[Notice that we had to include the possibility of equality 
since even though a,b,c,d are all distinct, that does not
guarantee that 
 and 
 are distinct, e.g. if a=1,b=6,c=2,d=3, they are not distinct.]

So we have proved:



which is what we had to proved.

---------------------------------------

For different positive a,b,c, we need to prove



Sorry, I haven't figured out how to do this one yet.

------------------------------------

The generalization.  If all ai are 
positive numbers, then



and equality holds only when all the ai are equal.

Edwin