# Thread: Proving |a+b| = |a|+|b|

1. ## Proving |a+b| = |a|+|b|

Hi guys, I'm doing a case by case proof of

$|a+b| \leq |a|+|b|$

from Spivak, and he says:

When $a \geq 0$ and $b \leq 0$, we must prove that:

$|a+b| \leq a-b$

I'm a bit stuck on his line of reasoning, could someone explain why we have to prove the above?

2. Originally Posted by rowe
I'm doing a case by case proof of
$|a+b| = |a|+|b|$ from Spivak, and he says
You cannot prove $|a+b|=|a|+|b|$ because it is not true.
So exactly what is the correct statement of this problem?

3. Typo, sorry. Fixed original post

4. Originally Posted by rowe
Hi guys, I'm doing a case by case proof of

$|a+b| \leq |a|+|b|$

from Spivak, and he says:

When $a \geq 0$ and $b \leq 0$, we must prove that:

$|a+b| \leq a-b$

I'm a bit stuck on his line of reasoning, could someone explain why we have to prove the above?
Hi

If $a \geq 0$ then $|a| = a$
and if $b \leq 0$ then $|b| = -b$

Therefore when $a \geq 0$ and $b \leq 0$, $|a|+|b| = a-b$