Expert: Paul Klarreich - 6/28/2009

Question
how do i prove this.    |x + y| <= |x| + |y|

Answer
Questioner:   Alan
Country:  Canada
Category:  Advanced Math
Private:  Yes  << don't mark private; I change it anyway.
 
Subject:  proofs
Question:  how do i prove this.    |x + y| <= |x| + |y|
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Hi, Alan,

Assuming x and y are numbers (not vectors or other things), just consider the cases:

A.  x,y both > 0,
Then  |x| = x, |y| = y, and |x + y| = |x| + |y|  

B. Suppose one of x,y = 0. Assume wlog, that it is y. Then:

|x + y| = |x| = |x| + |y|

C1. Suppose  x > 0, y < 0  and that  |x| > |y|
   Then  | x + y | = | |x| - (-y) | = | |x| - |y| | = |x| - |y|,
   because |x| > |y|
and  |x| - |y| < |x| + |y|  because  - |y| < |y|

You can look at the rest of the cases yourself.  Keep in mind that if  a<0. then  |a| = -a  ("the opposite of a")