Problem A. 624. (October 2014)
A. 624. \(\displaystyle a)\) Prove that for every infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) there exists some \(\displaystyle C>0\) such that for every positive integer \(\displaystyle r\) there are positive integers \(\displaystyle n\), \(\displaystyle m\) satisfying \(\displaystyle |n-m|\ge r\) and \(\displaystyle |x_n-x_m|<\frac{C}{|n-m|}\).
\(\displaystyle b)\) Show that for every \(\displaystyle C>0\) there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) and a positive integer \(\displaystyle r\) such that \(\displaystyle |x_n-x_m|>\frac{C}{|n-m|}\) holds true for every pair \(\displaystyle n\), \(\displaystyle m\) of positive integers with \(\displaystyle |n-m|\ge r\).
(CIIM6, Costa Rica)
(5 pont)
Deadline expired on November 10, 2014.
Statistics:
5 students sent a solution. 5 points: Williams Kada. 2 points: 2 students. 0 point: 2 students.
Problems in Mathematics of KöMaL, October 2014