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Miklós Schweitzer 1961- Problem 6

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December 1, 2015
college contestsreal analysis

Problem Statement

6. Consider a sequence {an}n=1∞\{ a_n \}_{n=1}^{\infty} such that, for any convergent subsequence {ank}\{ a_{n_k} \} of {an}\{a_n\}, the sequence {ank+1}\{ a_{n_k +1} \} also is convergent and has the same limit as {ank}\{ a_{n_k}\}. Prove that the sequence {an}\{ a_n \} is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence xn→∞ x_n \to \infty or −∞-\infty is considered to be convergente, too) (S. 13)
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