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Problem: $b_n$ is a bounded sequence and $a_n$ converges to 0. Prove that $a_n b_n -> 0$.

I would understand how to do this proof if I knew that the sequence $b_n$ was convergent. But that is not the case in general? Is there a different method to prove this?

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Suppose $|b_n|< K$. Given $\varepsilon>0,$ you can find $N\in\Bbb N$ (why?) such that $n>N$ implies $|a_n|<\dfrac{\varepsilon}{K}$, so $|a_nb_n|\ldots$

  • I see that $|b_n|<K$, but that does not imply convergence, so we cannot combine a and b? –  Jan 19 '14 at 04:32
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    In order to prove that the sequence $(a_nb_n)$ converges to $\ell$, you must show that for all $\varepsilon>0$ there exists $M\in \Bbb N$ such that $|a_nb_n-\ell|<\varepsilon$ if $n>M$. You know that definition, don't you? –  Jan 19 '14 at 04:35
  • Figured it out! Thank you. –  Jan 19 '14 at 04:39
  • You're welcome. –  Jan 19 '14 at 04:40