Absolute value theorem that I can't convince myself of

[JennyInTheSky]

While reading my text, I came across an inequality that I couldn't convince myself of...

For real numbers a,b: [tex]\left|a+b[/tex]|<= |a|+|b|. Is this something proven? Or is it an axiom or something?

 

[dodo]

Here,

http://en.wikipedia.org/wiki/Triangle_inequality

 

[PowerIso]

It's a pretty important inequality. I highly suggest you convince yourself of its truth :D

 

[gunch]

To convince yourself of its truth consider what the effect of the signs of a and b have on the inequality.

Case 1( a and b are positive):
|a+b| = a + b = |a|+|b|

Case 2 (a is positive and b is non-positive):
Let b = -y then a and y are positive. If a-y is positive:
[tex]|a+b|=|a-y| = a-y \leq a \leq |a| + |b|[/tex]
If a-y is non-positive, then y-a is positive and:
[tex]|a+b|=|a-y| = y-a \leq y = |b| \leq |a| + |b|[/tex]

Case 3 (a and b are negative):
Let a = -x, b = -y:
|a+b| = |-(x+y)| = x+y = |a|+|b|