I need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$.
Case 1: $x > 0$ and $y > 0$: the inequality simplifies to: $|x-y|\le |x-y|$ and we are done this case
Case 2: $x < 0$ and $y < 0$: the inequality simplifies to: $|-x + y| \le |x - y|$. Here we let $z = y-x$ and we see $|z| = |-z|$ and we are done this case
Could somebody help me out with the last two cases and provide a detailed explanation? I have trouble "splitting up" the cases.