Proving $\lfloor\frac{b}{a}(u+0.5)\rfloor-\lceil\frac{b}{a}(u-0.5)\rceil-\lfloor\frac{b}{a}(-u+0.5)\rfloor+\lceil\frac{b}{a}(-u-0.5)\rceil=0$

Question

I'm struggling proving $P(u)=0$ for all integer $u$, where $a$, $b$ are integers and

$$ P(u) = \left\lfloor\frac{b}{a}\left(u+\frac12\right) \right\rfloor - \left\lceil \frac{b}{a}\left(u-\frac12\right) \right\rceil - \left\lfloor\frac{b}{a}\left(-u+\frac12\right)\right\rfloor + \left\lceil\frac{b}{a}\left(-u-\frac12\right)\right\rceil$$

I've tried to let some part as integer plus alpha thing or mathematical induction but couldn't prove it.

Help me.