Hi I am working on my homework for Discrete Math and I don't know how to do this proof. It says, "Use proof by cases that |x+y|≤ |x| + |y| for all real numbers x and y." I know what proof by cases is but I don't know where to start.

Ok....I have started doing that and this is what I have so far:

Case 1: x + y>0

|x+y|>0 → |x+y| = x + y.

x + y cannot be > |x| + |y|.

∴ x + y ≤ |x| + |y|

Case 2: x + y = 0

|x + y| = 0 → x=x & y= -x

∴ x + (-x) ≤ |x| + |y|

Case 3: x + y < 0

|x + y| < 0 → x + y = -(x + y).

I have gotten this far, but I don’t know where to go from here. I know it makes sense in my head, but I don’t know how to put it into words.