Let $(a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\subseteq\mathbb R$. Moreover assume that $\lim_{n\to \infty} a_n=c_1\in\mathbb R$ with $c_1\neq 0$ and that $\lim_{n\to \infty} a_nb_n=c$ for some $c\in\mathbb R$.
Then I was asking myself if under the upper assumptions one can conlude that there exists some $c_0\in \mathbb R$ such that $b_n$ converges to $c_0$.
I composed a proof to verify the statement.
The proof is,
Since $c_1\neq 0$ we can wlog assume that there exists some $N\in\mathbb N$ such that $\forall n\geq N:\; a_m>0$. Hence for $m\geq N$ we have $a_nb_n=c\quad\Leftrightarrow \quad b_n=\frac{c}{a_n}$ and consequently $\lim_{n\to \infty}b_n=\lim_{n\to \infty}\frac{c}{a_n}=\frac{c}{c_1}$ i.e $b_n$ is convergent.
Since I somehow can not remember that I saw this statement in my basic Analysis courses I somehow doubt that it is true... Is there a mistake in my proof? I Also doubt that it is true because if someone relaxes the assumption that $c_1\neq 0$ and allows $c_1$ to be zero then for every bounded sequence $b_n$ we have $a_nb_n=0$ and in particular this holds also for not converging bounded sequences.
Thanks in advance!
