For $A \geq B$, both are strictly positive integers, is the following true? $$A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq \lfloor A/B \rfloor \times (B+1)$$
I tried the technique used in proving a very similar question: Prove/Disprove: $A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq (\lfloor A/B \rfloor + 1) \times B$ for $A \geq B$
But it seems like it didn't work in proving this. I also tried empirically generating random A and B's, but also can't find a counterexample.