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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 10 November 2014.


Statistics:

5 students sent a solution.
5 points:Williams Kada.
2 points:2 students.
0 point:2 students.

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