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Let $a_n$ be a null sequence and let $b_n$ be a bounded sequence. Prove that $a_n \cdot b_n$ is a null sequence.

I tried using the product rule of sequences but cannot because $b_n$ is not necessarily convergent and may not have a limit. How do I go about answering this?

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    $\begingroup$ The title isn't supposed to replace the first line of your question. Do you mean with "$a$ being a null sequence" that "$a$ converges to $0$"? $\endgroup$
    – Git Gud
    Oct 25, 2014 at 16:26
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    $\begingroup$ Use the fact that $b_n$ is bounded, and squeeze theorem $\endgroup$
    – FormerMath
    Oct 25, 2014 at 16:26
  • $\begingroup$ @GitGud Null sequence seems to mean sequence convergent to $0$. $\endgroup$
    – leo
    Oct 25, 2014 at 16:32

2 Answers 2

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Let $\varepsilon>0$ be given.

Since the sequence $\{b_n\}$ is bounded, there is a positive number $M$ so that $|b_n| \leq M$ for all $n$.

Since $\underset{n \to \infty}\lim a_n = 0$, there is a positive integer $N$ so that $|a_n|<\varepsilon/M$ whenever $n \geq N$.

Hence $|a_n \cdot b_n|<\varepsilon$ whenever $n \geq N$.

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If $|b_n| \leq M$ for all $n$, then $|a_nb_n| \leq |a_n|M$ for all $n$.

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