You can write it like this
Now, you set x=u+v and y=-v
using that |-v|=|v|
|u| - |v| |u+v|
what is you want to prove.
I'm trying to do some self-studying to refresh and improve my maths skills but I'm struggling with an exercise to provide a proof in a specific way:
Noting that x=x+y-y, apply the theorem |x+y|<=|x|+|y| together with the fact that |-y|=|y| to prove that |x+y|>=|x|-|y|
I can see intuitively that it's true, but I haven't quite got the knack of manipulating magnitude inequalities. Any help would be much appreciated.