I'm not an expert and I haven't dealt with these kinds of inequalities for a while, so I'm hoping someone can explain if my thinking 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, take the positive case: then we have that $|x+y| - |x| - b \leq |y| - b$.
- For the negative case, we have that $|x| - |x+y| +b \geq b - |y|$.
So, does this then mean that$$b - |y| \leq | |x+y| - |x| - b| \leq |y| - b$$ which in turn implies, after combining the cases, that$$| |x+y| - |x| - b| \leq | |y| - b|\;?$$
I sure hope so! Thanks in advance for answering.