# Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

How do I prove:

Let $\lim\limits_{n\to \infty} a_n = \infty$ and $\lim\limits_{n\to \infty} b_n = \infty$

Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

Thank you

• do you at least feel that this should be true somehow? – user87543 Nov 1 '13 at 10:37

Notice can find and $\alpha$ such that $0 < \alpha < \lim b_n$. For large $n$, in particular, $b_n > \alpha.$(proof?)
Let $M > 0$ . Therefore, by hypothesis we can find $N$ such that $a_n > \frac{M}{\alpha}$ for all $n > N$
$$a_nb_n > \alpha \frac{M}{ \alpha} > M$$
$$\therefore (a_nb_n) \to \infty$$
Hint: For $n$ large enough, $b_n \geq 1$ and $a_n >0$ hence $a_n \cdot b_n \geq a_n$.