Problem A. 739. (December 2018)

A. 739. Let \(\displaystyle a_1,a_2,\ldots\) be a sequence of real numbers from the interval \(\displaystyle [0,1]\). Prove that there is a sequence \(\displaystyle 1\le n_1<n_2<\ldots\) of positive integers such that

\(\displaystyle A=\lim_{\substack{i, j\to \infty\\i\ne j}} a_{n_i+n_j} \)

exists, i.e., for every real number \(\displaystyle \varepsilon>0\) there is a \(\displaystyle N_\varepsilon\) such that \(\displaystyle \big|a_{n_i+n_j}-A\big| < \varepsilon\) is satisfied for any pair of distinct indices \(\displaystyle i,j>N_\varepsilon\).

CIIM 10, Colombia

(7 pont)

Deadline expired on January 10, 2019.

Statistics:

4 students sent a solution.
7 points:	Pooya Esmaeil Akhoondy, Schrettner Jakab.
1 point:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, December 2018