MathDB

For a real number

x

, let

\|x \|

denote the distance between

x

and the closest integer. Let 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ , and let

\varepsilon >0

. Show that there exist infinitely many pairs

(n,m)

of indices such that n \not\equal{} m and \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right). V. T. Sos

Problem Statement