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sequence with countably infinite distinct subsequences

Source: LIMIT 2020 Cat 2 Objective P20

June 9, 2020
SequencesCountableConvergencecalculus

Problem Statement

Let {an}n\{a_n \}_n be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) {an}n\{a_n \}_n is bounded (B) {an}n\{a_n \}_n is unbounded (C) The set of convergent subsequence {an}n\{a_n \}_n is countable (D) None of these
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