0\leq\lfloor x\rfloor\{x\}+\{x\}^2-\lfloor x\rfloor\leq\lfloor x\rfloor\frac{\lfloor x\rfloor-2}{\lfloor x\rfloor-1}+\left(\frac{\lfloor x\rfloor-2}{\lfloor x\rfloor-1}\right)^2-\lfloor x\rfloor=\frac{4-3\lfloor x\rfloor}{(\lfloor x\rfloor-1)^2}