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In my assigment I have a question, the following:

Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $

Prove if the following is true or false:

$if$ $\lim\limits_{n \to \infty}({b_n})=1 $ $then$ $\lim\limits_{n \to \infty}({a_n})=0$

I have a feeling it's false, but I can't find an example that contradicts it.

Thank you.

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Answer is yes.

Suppose that $a_nb_n\to 0$ and $b_n\to 1$. Choose $\varepsilon>0$. Since $a_nb_n\to 0$, there is a $n_1\in \Bbb N$ such that $|a_nb_n|<\frac{\varepsilon}{2}$, for all $n\ge n_1$. Since $b_n\to 1$, there is a $n_2\in\Bbb N$ such that $|b_n-1|<\frac{1}{2}$, for all $n\ge n_2$, i.e. $\frac{1}{2}<b_n<\frac{3}{2}$, for all $n\ge n_2$. Choose $n_0=\max\{n_1,n_2\}$. Then for all $n\ge n_0$, we have $$|a_n|=\left|\frac{1}{b_n}(a_nb_n)\right|<2|a_nb_n|<2\cdot\frac{\varepsilon}{2}=\varepsilon.$$ Hence $a_n\to 0$.

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$$0=\lim_{n\rightarrow\infty}a_{n}b_{n}=\lim_{n\rightarrow\infty}a_{n}\lim_{n\rightarrow\infty}b_{n}=\lim_{n\rightarrow\infty}a_{n}.$$Note that the series are convergent, because if $$\lim_{n\rightarrow\infty}\left|a_{n}\right|=\infty $$ we have necessary $$\lim_{n\rightarrow\infty}b_{n}=0$$ which contradicts our hypothesis.

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  • $\begingroup$ Thanks, but I can't use limits arithmetic because no one says the sequences are Convergent. $\endgroup$ – Alan Mar 25 '15 at 20:40
  • $\begingroup$ @Alan I expand my answer. $\endgroup$ – Marco Cantarini Mar 25 '15 at 20:48

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