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Suppose I have two sequences, $a_n$ and $b_n$. I know that:

$\lim_{n\to\infty} a_n=1$ and that $\lim_{n\to\infty} a_nb_n=c$.

Does this mean that $\lim_{n\to\infty} b_n$ converges?

If so, by algebra of limits does it mean that $\lim_{n\to\infty} b_n=c$?

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Yes, you are right. – Michael Greinecker Nov 6 '13 at 0:43
Any ideas on a proof? – adrug Nov 6 '13 at 0:44
up vote 0 down vote accepted

Yes it does. We have that $$\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{a_nb_n}{a_n}=\frac{\lim_{n\rightarrow \infty}a_nb_n}{\lim_{n\rightarrow \infty}a_n}=\frac{c}{1}=c,$$ since the limit of a quotient is the quotient of the limits if they exist and the last is not $0$.

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Thanks, this works. Last question here - what is the Tex syntax you used for the n->inf UNDER the limit, not next to it? It looks so much clearer! – adrug Nov 6 '13 at 0:49
The same you used, but when you have a separated equation, the subscript for the $\lim$ is under it. – Mateus Sampaio Nov 6 '13 at 0:51

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