Proof that when $\lim_{n\to +\infty} a_n=+\infty$,$\lim_{n \to +\infty} b_n=-\infty$, then $\lim_{n \to +\infty}a_nb_n=-\infty$. This is a calculus homework question, I know how to prove if $ \lim_{n\to +\infty} a_n=A$ and $\lim_{n\to -\infty}=B$ then $\lim_{n\to+\infty}a_nb_n=AB$. But does $+\infty \cdot -\infty$ really make sense?
I'm kind of stuck. Would appreciate any help :)
Thanks in advance.
 A: Hint: Take $M<0$. If $n$ is large enough, then $a_n>\sqrt{-M}$ and $b_n<-\sqrt{-M}$. What does this tell you about $a_nb_n$ if $n$ is large enough?
A: Well, you should just write down the definitions...


*

*$\lim a_n b_n = -\infty$ if
$$
\forall_{M>0} \exists_p: n > p \Rightarrow a_n b_n < -M
$$

*$\lim a_n = +\infty$ if
$$
\forall_{M>0} \exists_p: n > p \Rightarrow a_n >M
$$

*$\lim b_n = -\infty$ if
$$
\forall_{M>0} \exists_p: n > p \Rightarrow b_n < -M
$$
So, just fix some arbitrary $M>0$ and, considering 2. and 3. choose $p_1, p_2$ such that  $a_n > \sqrt{M}$ (for all $n>p_1$) and $b_n < -\sqrt{M}$ (for all $n>p_2$). Now you can see that for $p \ge \max\{p_1, p_2\}$, definition  1. is satisfied.
A: Yes, it does make sense. It is defined operation.
$$ (+∞) \cdot (-∞) = -∞$$
For your case if $$\lim_{n \to +∞} a_n= +∞ \ and \lim_{n \to +∞} b_n=-∞,$$
then $$\lim_{n \to +∞} a_n \cdot b_n= -∞.$$
A: By definition we have $\forall M_1,M_2$


*

*$\forall n\ge n_1\quad a_n\ge |M_1|$

*$\forall n\ge n_2\quad b_n\le -|M_2|$
therefore by $n_0=\max\{n_1,n_2\}$ and $|M|=|M_1M_2|$


*

*$\forall n\ge n_2\quad a_nb_n\le -|M|$
