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Show that if ${a_n}$ and ${b_n}$ are sequences for which $\displaystyle\lim_{n\to\infty}a_n=0$ and ${b_n}$ is bounded, then $\displaystyle\lim_{n\to\infty}a_nb_n=0$

Sorry I'm on my phone and I'm not sure how to do all the correct notation. My idea was to use the theorem that says $\displaystyle\lim_{n\to\infty}a_nb_n=\left(\displaystyle\lim_{n\to\infty}a_n\right)\left(\displaystyle\lim_{n\to\infty}b_n\right)=0\cdot\left(\displaystyle\lim_{n\to\infty}b_n\right)$. But I have no idea how to prove this theorem and I am not sure how to incorporate the fact that b I bounded.

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    $\begingroup$ Your approach is wrong because the limit $\displaystyle\lim_{n\to\infty}b_n$ may not exist. Take for example $b_n=(-1)^n$. $\endgroup$ – user 170039 May 17 '15 at 6:11
  • $\begingroup$ Does bn being bounded have anything to do with this? Also instead of just telling me its wrong could you possibly give me hint because I am so stuck. $\endgroup$ – Felicia May 17 '15 at 7:13
  • $\begingroup$ Since $\displaystyle\lim_{n\to\infty}a_n=0$ therefore for all $\varepsilon>0$ we have $\left\lvert a_n\right\rvert<\frac{\varepsilon}{K}$ for all sufficiently large $n$ where $K$ is the upperbound of $\left\lvert b_n\right\rvert$ for all $n$. Now use the hint below. $\endgroup$ – user 170039 May 17 '15 at 9:02
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hint: $0\leq |a_nb_n| \leq K|a_n|, n \geq N_0$.

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  • $\begingroup$ Not sure how this helps sorry. $\endgroup$ – Felicia May 17 '15 at 7:14

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