Diophantine approximations with Fibonacci numbers

Let $F_n$ be the $n$-th Fibonacci number. Put $\phi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$:

1. ${\inf }_{n\in \mathbb{N}}\mathrm{\mid }\mathrm{\mid }{F}_n\alpha \mathrm{\mid }\mathrm{\mid }\le \frac{\phi -1}{\phi +2}$,
2. ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\mathrm{\mid }\mathrm{\mid }{F}_n\alpha \mathrm{\mid }\mathrm{\mid }\le 1\mathrm{/}5$,
3. ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\mathrm{\mid }\mathrm{\mid }{\phi }^n\alpha \mathrm{\mid }\mathrm{\mid }\le 1\mathrm{/}5$. These results are the best possible.