IOM 2019, Блиц — Задача 4

OLYMPIAD OF METROPOLISES, Moscow, September 2, 2019 · 2 септември 2019 г.

Условие

Assume $\lim\limits_{n\to\infty} a_n = A$, $b_n = |a_n - A|$. Which of the following statements are correct?

1. $\{b_n\}$ is a non-increasing sequence.

2. There exists a positive integer $N$ such that $\{b_N, b_{N+1}, \ldots\}$ is a non-increasing sequence.

3. There exists a positive integer $N$ such that $b_n = 0$ for any $n \ge N$.

4. There exists a sequence of positive integers $k_1 < k_2 < \ldots$ such that $\{b_{k_1}, b_{k_2}, \ldots\}$ is a non-increasing sequence.

Решение

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Оригинал в Архива: blitz-en-all-task-final.pdf