True or false (prove if true):
- If $\{a_n\}$ converges to $l\neq 0$ and $\{b_n\}$ diverges, then $\{a_nb_n\}$ diverges.
- If $\{\frac{a_n}{b_n}\}$ converges to an irrational number $l$, then that $|b_n|\rightarrow \infty$ as $n\rightarrow \infty$.
Attempt:
I know if $a_n=1/n$ and $b_n=n$ then $\{a_nb_n\}$ converges to 0. But here, $a_n\rightarrow l\neq 0$. Please provide an answer.
- $\{\frac{a_n}{b_n}\}$ converges to an irrational number $l$ then $$|\frac{a_n}{b_n}-l|<\epsilon$$ i.e $$|a_n-lb_n|<\epsilon |b_n|$$. Nothing is given about $\{a_n\}$. I tried to prove by contradiction by assuming $b_n$ is bounded but fails. Please help.