Prove the triangle inequality I want to porve the triangle inequality:
$x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$
I figured out that probably the cases:


*

*$x\geq0$ and $y \geq 0$

*$x<0$ and $y < 0$

*$x\geq0$ and $y < 0$

*$x<0$ and $y \geq 0$ <- Here I am not sure...


have to be proven. However, I do not figured out a concrete method. Are my assumptions true?  How to finish the prove with these assumptions. I really appreciate your answer!!!

You can use the following 

For any $x,a\in\Bbb R$, we have that  $|x|\leq a$ if and only if $-a\leq x\leq a$

Then since we have $-|x|\leq x\leq |x|$ and $-|y|\leq y\leq |y|$ for any pair $x,y$, this gives $$-(|x|+|y|)\leq x+y\leq |x|+|y|$$
which by the claim is equivalent to $|x+y|\leq |x|+|y|$, which is the triangle inequality.
