Is the title's statement true?
I was able to prove that $-\sup(\vert B \vert-\vert A \vert) \leq \sup \vert A \vert - \sup \vert B \vert \leq \sup(\vert A \vert - \vert B \vert)$
and of course that $\sup \vert \vert A \vert - \vert B \vert \vert = \max \{ \inf(\vert A \vert - \vert B \vert) , \sup(\vert A \vert - \vert B \vert)\}$.
and from that the statement is obvious, but I'm still not quite sure if I have not done a terrible mistake in my profs. ( it was very confusing :) )
so would like to know if it is really true.