丢番图逼近

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对于任意无理数$x$，证明：存在无穷多个正整数$n$,使得$\{nx\}<\frac 1n$

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Ivan Niven, Diophantine Approximation (Theorem 3.1, page 23)
Given an irrational number $\alpha$ there is an increasing sequence of positive integers: ${k_1}<{k_2}<\cdots$ such that for $i\geq1$, $$\{{{\alpha}k_i}\} < {\frac{1}{k_{i+1}}}$$