This was proved by Chung and Graham \cite{ChGr84}, who in fact prove that
$$\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq \frac{1}{c}\approx 0.3944$$
where
$$c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.5353705\cdots$$
and $F_m$ is the $m$th Fibonacci number.
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