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 ε>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 combinatoricsinequalitiesreal analysiscollege contestsalgebra