Cauchy convergent sequences

Question

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample to show why it is not true.

My approach: let $lim(a_n)$ = $L$ and $\lim(b_n)$ = $M$, with $M \neq 0$. Since $(a_n)$ and $(b_n)$ converge, then by the property of limits $(a_n / b_n)$ will converge too with $M \neq 0$.

I don't know how to make it so that it is Cauchy.

Answer 1

The limit law you've cited is correct when $M \ne 0$, so try a case when $M = 0$. In particular, take $a_n$ to be a constant sequence (e.g. $a_n = 1$ for all $n$) and $b_n = \frac{1}{n}$, perhaps. Then

$$\frac{a_n}{b_n} = \frac{1}{\frac{1}{n}} = n$$

is an unbounded, and thus divergent sequence.

Answer 2

Put $a_n = 1/\sqrt{n}$ and $b_n = 1/n$. You have $a_n/b_n = \sqrt{n}$.

Answer 3

Not necessarily. Take for example, $a_n=\frac{(-1)^n}{n}$, and $b_n=\frac{1}{n}$. Then $a_n/b_n=(-1)^n$, which is not Cauchy.