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!!!

If both $x$ and $y$ are $0$  or $x=-y$ then the inequality is clear. Otherwise we note that for $x,y\in\mathbb{R}$ $x\le|x|$ and similarly $y\le|y|$, which follows from the definition of the absolute value.
This tells us that $x+y\le|x|+|y|$ which implies $\frac{x+y}{|x|+|y|}\le1$ since $|x|+|y|>0$.
Thus, $|x+y|=|\frac{x+y}{|x|+|y|}|(|x|+|y|)=\frac{1}{sgn\bigg(\frac{x+y}{|x|+|y|}\bigg)}\frac{x+y}{|x|+|y|}(|x|+|y|)\le|x|+|y|$.
