# Inverse theorem on product of two convergent sequences

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

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 $0$.
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