Which of the following is sufficient for $\lim_{n\to\infty} \frac{a_n}{b_n} = 1$

Question

Assume both $a_n$ and $b_n$ are convergent. Which of the following is sufficient for $\lim_{n\to\infty} \frac{a_n}{b_n} = 1$

I. $\lim a_n = \lim b_n$

II. $\lim \frac{a_n ^2}{b_n^2} = 1$

III. $\lim \frac{b_n}{a_n} = 1$

IV. $\lim |\frac{a_n}{b_n}| = 1$

V. $|a_n - \frac{a_n}{n}| \le |b_n| \le |a_n|$

I know that I and IV are wrong since I have found counterexamples. V is wrong as well.

I am not sure about $II$ and $III$. I am tempted to say that $III$ is true because we can do something like $\frac{a_n}{b_n} = \frac{1}{\frac{b_n}{a_n}}$. But I couldn't come up with a counterexample for $II$.

Answer 1

For II: $a_n=-1$, $b_n=1$ gives a counterexample.

Answer 2

If $a_n=-1$ and $b_n=1$ for all $n$ then both sequences are convergent and $$\lim\frac{a_n^2}{b_n^2}=1$$ but $$\lim\frac{a_n}{b_n}=-1$$

The condition III is good, for the reason you said.