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