In 1920 H. Steinhaus (H. Steinhaus, Sur les distances des points de mesure positive, Fund. Math. 1 (1920) 93-104) proved the following result: ‘Let A be a Lebesgue measurable set of a positive measure. Then in A exist at least two points such that distance between them is a rational number.’
In this paper (talk) we shall proved that in A exists sequences (xn) of different points such that distance between any of its members is a rational number.