Prove: $|x|-|y|\leq |x-y|$

Question

Can you help me proving this equality? $$|x|-|y|\leq |x-y|$$

I think that it's possible to give a too short proof if the things are written correctly.

Answer 1

Hint: $$|x|=|(x-y)+y|$$ Now apply the triangle inequality.

PS: by symmetry you conclude that on the left hand side, you can even put an absolute value around the difference.

Answer 2

$$|x|-|y|\leq |x-y| \Leftrightarrow |x|\le|y|+ |x-y|$$ Use $|a|+|b|\ge|a+b|$ $$|y|+ |x-y|\ge |y+ x-y|=|x|$$