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So I am given two sequences, $(a_n)$ and $(b_n)$ with limits $\lim(a_n)=0$, $\lim(b_n)=+\infty$ but the limit of the product should equal $2$.

I am confused here because I was in belief that the product of a zero-sequence and any sequence should equal a zero-sequence and this if $(c_n)=(a_n \cdot b_n) \Rightarrow \lim(c_n)=\lim(a_n \cdot b_n)$. So my question is: Is there any example of two sequences $a$ and $b$ such that it is possible to have the limit of the product = $2$?

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$a_n=\frac{2}{n}$ and $b_n=n$ works. It's true that $\lim a_nb_n=(\lim a_n)(\lim b_n)$ provided that the limits of $\{a_n\}$ and $\{b_n\}$ exist and are finite; infinite limits have to be handled more carefully.

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