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And similarly is a unbounded sequence times a convergent sequence bounded? I'm still getting familiar with the properties of sequences.

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2 Answers

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An unbounded sequence times a sequence that has at least one nonzero convergence point is actually always unbounded.

That should answer both your questions. For a concrete counterexample, take any unbounded sequence and the bounded sequence $1,1,1,\dots$

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A counter example shows both: $a_n = 2$ is bounded, and $b_n = n$ is unbounded.But $a_nb_n = 2n$ which is unbounded, so we don't have the first. Also $a_n = 2$ converges to $2$, so it covers your second part as well.

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