lean_workbook_plus_56751

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