All the arguments are perfect, and the proofs can't be improved. Observe that the properties remain true if you replace complex numbers and modulus by vectors and norms in any other normed space.
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.