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I have a basic question about the limits of rational sequences:

Suppose we know that a sequence $(a_{n})$ converges to $0$ and that each $a_n$ is defined by $a_n := \frac{b_n}{c_n}$ for two sequences $(b_{n})$ and $(c_{n})$. We further know that $(c_{n})$ goes to infinity, i.e

$$\lim_{n \rightarrow \infty} c_n = \infty.$$

Can we deduce from this that $\lim_{n \rightarrow \infty} \ b_n = 0$ and if yes how?

Edit: I have an additional question: Are we at least able to deduce that $b_{(n)}$ converges?

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No.

What about $$\left( a_n \right)=\left( \frac{1}{n}\right)$$

where $a_n=\frac{1}{n}$, $ b_n=1$ and $ c_n=n$

Edit-Edited to answer your additional question.

Again No.

What about $$\left( a_n \right)=\left( \frac{n}{n^2}\right)$$

where $a_n=\frac{n}{n^2}$, $ b_n=n$ and $ c_n=n^2$

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No, that’s not true. For instance, for $b_n = n$ and $c_n = n^2$, we have

$$\lim_{n \to \infty}\frac{b_n}{c_n} = 0$$

while both sequences tend to $\infty$. More generally, when both $b_n$ and $c_n$ are polynomials with $c_n$ having the higher degree, the limit tends to $0$, even though both polynomials may tend to $\infty$.

Edit: Also no, the example above once again shows this. Both polynomials clearly diverge to $\infty$, but since the denominator has a greater degree, it diverges faster. Hence, the limit tends to $0$.

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