ISL 1979 Problem 56

Show that for every $n\in\mathbb{N}$ , $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$ , there exists an $n\in\mathbb{N}$ such that $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$ .