I'm not an expert and I haven't delt with these kinds of inequalities for a while, so I'm hoping someone can explain if my thought process is reliable and will suffice to prove the results I'm looking for.
I start with the expression $| \ |x+y|-|x|-b \ |$, and I'd like to know if this is less than or equal to
$| |y| - b|.$
My approach is first, taking the positive case, we have $|x+y| - |x| - b \leq |y| - b$.
For the negative case, we have that |x| - |x+y| +b \geq b - |y|.
So, does this then mean
$b - |y| \leq | |x+y| - |x| - b| \leq |y| - b$ which then implies, after combining the cases, that
$| |x+y| - |x| - b| \leq | |y| - b|$?
I sure hope so! Thanks in advance for answering.