Suppose a, b, c are real numbers such that a+b \geq 0, b+c \geq 0, and c+a \geq 0. Prove that
a+b+c \geq \frac{|a|+|b|+|c|}{3}
(Note: |x| is called the absolute value of x and is defined as follows. If x \geq 0 then |x|=x; and if x<0 then |x|=-x. For example, |6|=6,|0|=0 and |-6|=6.)