1. Prove that given that a, b, and c are positive real numbers.
2. Prove that given that x, y, and z are positive real numbers.
expanding gives us:
therefore but again by (1): thus:
now in (1) put and then divide by to prove your first problem and in (2) put and then divide by to prove your second problem.
On the first inequality, we can start by using
(try expanding the left hand side to see that this is true).
But if x,y, and z are all positive, then xyz is positive. And as is the sum of three squares, it cannot be negative. Thus . From our above equality, then, we have
Also, I don't understand your solution. Could you please indicate which parts correspond to which question? Thanks!