Prove that for every two real numbers x and y, it is true that $\vert x\vert + \vert y\vert\ge\vert x - y\vert$. Show that there exist real numbers for which $\vert x\vert + \vert y\vert = \vert x - y\vert$.
I don't quite know the right way to prove this. Thanks in advance.
$|x|+|-y|\geq|x-y|$ by the triangle inequality, as already stated by @Don Thousand
$\vert x\vert + \vert y\vert = \vert x - y\vert$ is always the case for numbers with $x \ge -y \ge 0$, or the same with $x$ and $y$ exchanged.