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Given $\lim \limits_{n\to \infty}a_n=L$ and $\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$ then $\lim \limits_{n\to \infty}b_n=L$

I think that this claim is true:

$\lim \limits_{n\to \infty}a_n=L$

$\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$

using limits arithmetics I can say:

$\lim \limits_{n\to \infty}\frac{a_n}{b_n}=\frac{\lim \limits_{n\to \infty}a_n}{\lim \limits_{n\to \infty}b_n}=1$

$\frac{\lim \limits_{n\to \infty}a_n}{\lim \limits_{n\to \infty}b_n}=1$

$\lim \limits_{n\to \infty}b_n\cdot 1=\lim \limits_{n\to \infty}a_n$

$\lim \limits_{n\to \infty}b_n=L.$

is my proof correct?

2 Answers 2

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I'm not sure it's correct. In the first line after "I can say", it seems that you are assuming $\lim\limits_{n\rightarrow\infty}b_n=L$, which is what you are trying to prove.

However, you can use the following fact: Suppose $\lim\limits_{n\rightarrow\infty}a_n= a$ and that $\lim\limits_{n\rightarrow\infty}c_n= c$. Then if $c\ne0$, the limit of the quotient ${a_n\over c_n}$ as $n$ tends to infinity exists and $\lim\limits_{n\rightarrow\infty}{a_n\over c_n}={a\over c }$.

Apply this with $c_n={a_n\over b_n}$.

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    Awesome sauce, thanks a lot :)2012-04-01
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Given $\lim \limits_{n\to \infty}a_n=L$ and $\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$ then $\lim \limits_{n\to \infty}b_n=L$

Try $\lim \limits_{n\to \infty}\frac{a_n}{b_n}\frac1{a_n}=\frac1L$

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    They're not different, I posted mine before the other guy. It's all the same.2012-04-02