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!!!
 A: http://www.proofwiki.org/wiki/Triangle_Inequality/Real_Numbers  - real numbers
https://www.math.ucdavis.edu/~forehand/67_summer2012_files/triangle%20inequality.pdf  - complex numbers 
A: Hint: If both $x,y \geq 0$ or both $x,y \leq 0$ then we just have, 
$$|x+y|=|x|+|y|$$
If one is greater than zero and one is less than zero, then it should be obvious that if it does not matter which one I prove it for. 
A: My favorite way to deal with absolute values is to use the definition I learned from Edsger W. Dijkstra's writings probably learned from either Wim Feijen or Jeremy Weissmann, viz. $$|x| = x \:\max\: -x$$ (see Wim Feijen and Netty van Gasteren's WF228 [PDF] for the earliest reference I could find) and then use the nice properties of $\;\max\;$ instead of the less nice properties of $\;|\cdot|\;$.  Here this helps me a lot, since it prevents a proof by cases.
Working from the right of the left of the equation, the proof becomes a straightforward calculation:
\begin{align}
& |x|+|y| \\
= & \;\;\;\;\;\text{"the above definition of $\;|\cdot|\;$, twice"} \\
& (x \:\max\: -x) + (y \:\max\: -y) \\
= & \;\;\;\;\;\text{"$\;+\;$ distributes over $\;\max\;$, three times"} \\
& (x + y) \:\max\: (x-y) \:\max\: (-x+y) \:\max\: (-x-y) \\
= & \;\;\;\;\;\text{"reintroduce $\;|\cdot|\;$ using the above definition, twice"} \\
& |x+y| \:\max\: |x-y| \\
\geq & \;\;\;\;\;\text{"basic property of $\;\max\;$"} \\
& |x+y| \\
\end{align}
Note that in the middle steps I have implicitly used the fact that $\;\max\;$ is associative and symmetric.
A: 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, $|x+y|=|\frac{x+y}{|x|+|y|}|(|x|+|y|)=\frac{1}{sgn\bigg(\frac{x+y}{|x|+|y|}\bigg)}\frac{x+y}{|x|+|y|}(|x|+|y|)\le|x|+|y|$.
A: 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.
