prove that |a+b|<|a|+|b| is true for all reall numbers

Printable View

- Sep 9th 2007, 05:33 AMkeyinequality
prove that |a+b|

__<__|a|+|b| is true for all reall numbers - Sep 9th 2007, 06:22 AMThePerfectHacker
If $\displaystyle a\geq 0$ we end up with:

$\displaystyle |a+b|\leq a + |b|$.

Now if $\displaystyle -a\leq b <0$ we have:

$\displaystyle a+b\leq a + |b| \implies b\leq |b|$ which is true.

And if $\displaystyle 0\leq b$ we have:

$\displaystyle a+b\leq a+b$ which is true.

Now if $\displaystyle a<0$ let $\displaystyle a=-c$ where $\displaystyle c>0$ so:

$\displaystyle |a+b|\leq |a|+|b| \implies |c+(-b)| \leq |c|+|-b|$.

Which we established above already.