Subtract this inequality $-|y|\leq y \leq |y| $ , from this inequality $-|x|\leq x \leq |x|$ to get $-(|x|-|y|) \leq x-y \leq |x|-|y|$. Using the property $-a \leq x \leq a \implies |x| \leq a$, we obtain $|x-y| \leq |x|-|y|$.
However, the reverse triangle inequality has the inequality sign reversed. I don't understand what's wrong with my proof. Please help. Thank you.
You can't subtract an inequality from another and expect to get a valid inequality.
For example, $1<2$ and $3<4$ are valid,
but $3-1 \not < 4-2$.