0<\lfloor x\rfloor+1-\lfloor x\rfloor\{x\}-\{x\}^2\leq \lfloor x\rfloor+1-\lfloor x\rfloor\frac{-\lfloor x\rfloor}{1-\lfloor x\rfloor}-\left(\frac{-\lfloor x\rfloor}{1-\lfloor x\rfloor}\right)^2=\frac{1-\lfloor x\rfloor-\lfloor x\rfloor^2}{(\lfloor x\rfloor-1)^2}