I have to demonstrate that $|x+y| < \max(|x|, |y|) \Leftrightarrow xy < 0$. I'm bit lost as how to proceed on this. I know I have to separate in two cases and that the right side is $x$, when $x > 0$ and $y$ when $y > 0$ (only one of the two can be positive).
I can separate the cases $|x+y|>0$ ($x > 0$ and $x > |y|$, though the cases should be just $x > 0$ and $y > 0$, but this doesn't help much) and $|x+y|<0$ but I don't know what to do from there. So for the first case I have $x+y < x$, which is pretty obvious I guess, as $y$ is negative? Do I have to write anything else to prove this?
Thanks in advance!