For the following statements, prove or give a counterexample: a) If a convergent sequence is bounded, then it is monotone. b) Every convergent sequence can be represented as the sum of two oscillating sequences.
Added by Alfredo A.
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Counterexample: Consider the sequence \(a_n = (-1)^n\) for \(n \geq 1\). This sequence converges to 0 but is not monotone as it alternates between -1 and 1. However, it is bounded since \(-1 \leq a_n \leq 1\) for all \(n\). Show more…
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