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Let $F_n$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$: 1) ${inf}_{n\in \mathbb{N}}\left|\right|F_n\alpha \left|\right|\le \frac{\varphi -1}{\varphi +2}$, 2) ${lim\; inf}_{n\to \infty }\left|\right|F_n\alpha \left|\right|\le \frac{1}{5}$, 3) ${lim\; inf}_{n\to \infty }\left|\right|{\varphi }^n\alpha \left|\right|\le \frac{1}{5}$. These results are the best possible.