Prove/Disprove: $A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq (\lfloor A/B \rfloor + 1) \times B$ for $A \geq B$

I've been trying to prove the following, for $$A \geq B$$, both are strictly positive integers: $$A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq (\lfloor A/B \rfloor + 1) \times B$$ Not sure if it's true. Can't find a counterexample so far. Anyone has an idea?

$$A\ge B\implies\lfloor A/B \rfloor\ge 1$$ and $$\lceil A/B \rceil\ge 1$$ $$A-\lfloor A/B\rfloor - \lceil A/B \rceil
where the last inequality holds since $$\lceil A/B \rceil = \lfloor A/B \rfloor$$ if $$A$$ is divisible by $$B$$, otherwise $$A/B$$ is not an integer and $$\lceil A/B \rceil = \lfloor A/B \rfloor +1$$.