These are elementary inequalities. Will appreciate help in a rigorous proof of these.
Assume the field is of Complex Numbers
Assume following is true/known/proven
|x+y| <= |x| + |y|
(I would want to avoid any other thoerm, unless really required)
Q1. ||x|-|y|| <= |x-y|
|(x-y)+y| <= |x-y| + |y|
|x| - |y| <= |x-y|
Similarly I can show, |y| - |x| <= |x-y|
Does this imply ||x|-|y|| <= |x-y| ?
(I can use an argument where |x| = x for x>0 and -x for x<0 and x in Real to show this. But wondering if that is correct/needed?)
Q2. |x| - |y| <= |x+y|
In part one we showed |x| - |y| <= |x-y|
Put y=-y in this.
(I am fairly satisfied with this attempt.)
Request if you guys can please check these/provide better/cleaner proofs.