Question

For x, y ∈ R, prove that ||x| − |y|| ≤ |x − y|, (Hint: consider x = y + (x − y)).​

Answer 1

Using the triangular inequality (and the given hint!), we have

$|x|=|(x-y)+y|\leq|x-y|+|y|\implies|x|-|y|\leq|x-y|$

Similarly,

$|y|=|(y-x)+x|\leq|x-y|+|x|\implies|y|-|x|\leq|x-y|\implies -|x-y| \leq |x|-|y|$

We managed to bound the quantities in this fashion:

$-b\leq a\leq b \implies |a|<b$

And thus we have

$||x|-|y||\leq |x-y|$