Fractional parts of integral multiples of an irrational number

Question

Fix an irrational number $\alpha \in (0,1)$. Let $\{x\}$ denote the fractional part of the real number $x$. Consider the sequence $$\{{\{\alpha}n\}:n=1,2,\cdots \}.$$ This sequence is uniformly distributed in the unit interval. However, is it the case that $${\{\alpha}n\}\ge {\frac{1}{n}}$$ for sufficiently large positive integers $n$?