Prove that xy

Prove that |x+y| <_ |x| + |y| for all real numbers x and y

Solution

Prove that for all real numbers x and y, | x + y| | x| + | y| .
Answer: Part (b) above tells us that | x| x | x| and | y| y | y| .
Adding these together we get that | x| | y| x + y | x| + | y| . Using part
(a), this tells us that | x + y| | x| + | y| as required