$$x\in (0,1)\Rightarrow y=\frac{1}{x}\in (1,\infty)\Rightarrow \left\lfloor y\right\rfloor = \left\lfloor\frac{1}{x}\right\rfloor \in[1,\infty)\Rightarrow\frac12<\frac{\left\lfloor y\right\rfloor}{y}=x\cdot \left\lfloor \frac1x\right\rfloor$$
$$\Rightarrow$$
$$1<2\cdot\frac{\left\lfloor y\right\rfloor}{y}=2\cdot x\cdot \left\lfloor \frac1x\right\rfloor $$