Let $c\in \mathbb{R}$. Find two sequences $(a_n)_n$, $(b_n)_n \subset \mathbb{R}$ with:
(i) $\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=0 $ and $\lim\limits_{n\to\infty}a_nb_n=c$
My example:
$a_n:=n$ and $b_n:=\frac{1}{n}$
(ii) $a_n \neq 0 \neq b_n$ for all $n\in \mathbb{N}$, $\lim\limits_{n\to\infty}a_n=0=\lim\limits_{n\to\infty}b_n$ and $\lim\limits_{n\to\infty}\frac{a_n}{b_n}=c$
My example:
$a_n:=\frac{1}{n^2}$ and $b_n:=\frac{1}{n}$
(iii) $\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=-\infty$ and $\lim\limits_{n\to\infty}(a_n+b_n)=c$
My example:
$a_n=n$ und $b_n=-n$
Are these valid examples?