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|

Attempt:

|(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|

Attempt:

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.

Thanks