Let $\lfloor \cdot \rfloor $ be the floor function. Is it truth that $|\lfloor ax \rfloor -\lfloor ay \rfloor |\leq |\lfloor bx \rfloor -\lfloor by \rfloor |$ for any real numbers $x,y$ and positive numbers $a\leq b$ ?
By the graph of floor functions, i feel it is true, but i can't prove this. Does someone know the reference to see this is correct or not ?
The result does not hold. Try $a=0.5$, $b=0.8$, $x = 1.8$, $y=2.2$. Below you can see the graph of $$ g(x,y)=\left| \lfloor ax\rfloor - \lfloor ay\rfloor\right|-\left| \lfloor bx\rfloor - \lfloor by\rfloor\right| $$
for the mentioned values of $a,b$. In the blue portions the inequality is not valid.