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?